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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:
where = the inertial force, = the rate of strain change, mg_{c} = 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:
The following solution assumption is made concerning the time function:
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
which is of the form
The standard solution of this equation is:
The constants of integration, C_{1} and C_{2}, are determined by the initial and boundary conditions, and φ is a collection of the constants and is given by:
Therefore, the total solution is:
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, A_{c}, E, c, ω, v_{s}, and φ are replaced by θ, I, J, G, c_{θ}, ω_{θ}, v_{θ}, and η, respectively. The torsional equation of motion is:
This gives the solution as:
Constants C_{1} and C_{2} are based on the initial and boundary conditions, and η is a collection of the constants and is given by:
where
and
Nomenclature
A_{c} | = | cross-sectional area, L^{2}, in.^{2} |
c | = | axial damping coefficient, mL/t, lbf-ft/sec |
c_{θ} | = | torsional damping coefficient, mL/t, lbf-sec/rad |
C | = | constant of integration, various |
d_{i} | = | inner diameter, L, in. |
d_{o} | = | outer diameter, L, in. |
E | = | modulus of elasticity, m/Lt^{2}, psia |
g_{c} | = | gravitational constant, L/t^{2}, 32.174 ft/sec^{2} |
G | = | shear modulus, m/Lt^{2}, psia |
i | = | imaginary operator |
I | = | second moment of inertia, L^{4}, in.^{4} |
J | = | polar moment, L^{3}, in.^{3} |
m | = | mass, m, lbm |
m_{θ} | = | mass polar moment of inertia, mL, lbf-sec^{2} |
t | = | time, seconds |
T(t) | = | displacement function in terms of time, t |
u | = | displacement, L, in. |
U(x,t) | = | continuous displacement function, L, in. |
v_{θ} | = | 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/L^{3}, lbm/in.^{3} |
ω | = | frequency, 1/t, Hz |
ω_{θ} | = | twist natural frequency, 1/t, Hz |
φ | = | convenient coefficient, 1/L, 1/ft |
Noteworthy papers in OnePetro
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External links
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See also
Vibrating systems with multiple sources of motion
Differential calculus refresher
PEH:Mathematics_of_Vibrating_Systems