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# Continuous vibration systems

Analyzing the effects of vibration get more complex as more sources of motion are added. Sources of motion are modeled as degrees of freedom (DOFs). Systems with one degree of freedom and two or more degrees of freedom are discussed elsewhere. If one continues to add DOFs, the limit at an infinite DOF defines a continuous system. The result becomes a partial differential equation (PDE). The following is a brief description of the separation of variables method for solving a PDE. (See this refresher on differential calculus if needed.)

## Calculating motion in a continuous system

Fig. 1 shows a freebody diagram for axial and torsional systems. The axial system equations will be used to determine the solution of the equations of motion. Eq. 1 is the axial equation of motion: ....................(1)

where = the inertial force, = the rate of strain change, mgc = the static weight of the element, and = the force from viscous damping. This PDE, Eq. 1, can be solved using the separation of variables method. This is shown as: ....................(2)

The following solution assumption is made concerning the time function: ....................(3)

This equation is substituted back into the assumed solution, which then is appropriately differentiated and substituted back into the equation of motion. The equation becomes ....................(4)

which is of the form ....................(5)

The standard solution of this equation is: ....................(6)

The constants of integration, C1 and C2, are determined by the initial and boundary conditions, and φ is a collection of the constants and is given by: ....................(7)

Therefore, the total solution is: ....................(8)

The solution to the torsional equation of motion is derived similarly to the axial equation, with the substitution of the appropriate variables and noting that there is no initial strain from gravity. The variables u, m, Ac, E, c, ω, vs, and φ are replaced by θ, I, J, G, cθ, ωθ, vθ, and η, respectively. The torsional equation of motion is: ....................(9)

This gives the solution as: ....................(10)

Constants C1 and C2 are based on the initial and boundary conditions, and η is a collection of the constants and is given by: ....................(11)

where ....................(12)

and ....................(13)

## Nomenclature

 Ac = cross-sectional area, L2, in.2 c = axial damping coefficient, mL/t, lbf-ft/sec cθ = torsional damping coefficient, mL/t, lbf-sec/rad C = constant of integration, various di = inner diameter, L, in. do = outer diameter, L, in. E = modulus of elasticity, m/Lt2, psia gc = gravitational constant, L/t2, 32.174 ft/sec2 G = shear modulus, m/Lt2, psia i = imaginary operator I = second moment of inertia, L4, in.4 J = polar moment, L3, in.3 m = mass, m, lbm mθ = mass polar moment of inertia, mL, lbf-sec2 t = time, seconds T(t) = displacement function in terms of time, t u = displacement, L, in. U(x,t) = continuous displacement function, L, in. vθ = torsional sonic velocity, L/t, ft/sec x = displacement, L, in. X(x) = displacement function in terms of location x y = dependent variable, various η = convenient coefficient, 1/L, 1/ft θ = twist, rad ρ = density, m/L3, lbm/in.3 ω = frequency, 1/t, Hz ωθ = twist natural frequency, 1/t, Hz φ = convenient coefficient, 1/L, 1/ft

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