Author name: GULSHAN SHAFI

The natural frequency is the rate at which a system oscillates autonomously.The formula is given by: Where: k is the stiffness of the system. m is the mass of the system.

The modal damping equation relates the damping ratio to system parameters.The formula is given by: Where: ξ represents the modal damping ratio, c is the damping coefficient, m is the mass of the system, K is the stiffness of the system.

Distance from axis with same moment of inertia mass.The formula is given by: Where: k is the radius of gyration, I is the moment of inertia of the object about the axis of rotation, and m is the mass of the object.

The modal stiffness equation defines the stiffness of a structure.The formula is given by: Where: Km​ = Modal stiffness m = Mass of the structure or system ω = Angular frequency of the mode of vibration

Material failure predicted by comparing strain energy to critical limit.The formula is given by: Where: U is the strain energy density, σ is the stress applied to the material, ϵ is the corresponding strain.

Critical load causing structural instability in elastic columns predicted mathematically.The formula is given by: Where: P is the critical buckling load, E is the modulus of elasticity of the material, I is the moment of inertia of the cross-sectional area of the column, k is the effective length factor (depends on the end conditions of the

Thin plate behavior analyzed, accounting for transverse shear deformation.The formula is given by: Where: D is the flexural rigidity of the plate. ∇⁴ is the biharmonic operator. w is the deflection of the plate. q(x,y) is the applied transverse load per unit area.

Plate theory extending Euler-Bernoulli’s for thick plates analysis.The formula is given by: Where: D is the flexural rigidity of the plate. ∇⁴ is the biharmonic operator. w is the deflection of the plate. q is the applied transverse load.

Graphical method for visualizing stress components in materials analysis.The formula is given by: Where: σx​ and σy​ are the normal stresses in the x and y directions, respectively. τxy​ is the shear stress acting on the xy-plane.

Material failure predicted when maximum stress exceeds critical value.The formula is given by: Where: σ₁ is the maximum principal stress σ_x is the normal stress in the x-direction σ_y is the normal stress in the y-direction