Natural small vibrations of a flat viscoelastic spiral spring

2. Methods
2.1 Problem statement and solution method
The curve on which the center of gravity of the cross section of the rod lies is called an elastic line. In our case it is a flat curve. The coordinates of the points of the elastic line will be characterized by the arc length measured from its end.
Let us introduce a fixed coordinate system ξ,η,ζ with the beginning at the end O of the elastic line (Fig. 1). Let's direct the axis ξ along the normal, η - along the binormal, ζ - along the tangent to the elastic line at point s. In addition, at each point of the elastic line introduced the Frenet trihedron where the unit is directed along the normal, - along the binormal, - along the tangent to the elastic line at point s. Let - is the curvature of the curve. Then the unit vectors of the fixed coordinate system are related to the unit vectors by the relations:
, (1)
where - is the angle of rotation of the tangent to the elastic line,
(2)

Fig.1. Design scheme of a spiral spring
The own vibrations of a thin inextensible curved rod with displacements in its plane are described by the equations [14-16]:

(3)

In the case of fixed ends, the following boundary conditions apply [17-19]:

(4)
Vibrations of a rod fixed at the ends with displacements perpendicular to the plane of the curve satisfy the equations [20-22]



and boundary conditions [23-25]

(6)
The unknown functions in (3) and (5) are: 𝑉_𝑥, - shear forces, 𝑉_𝑧 - tensile force acting in the section of the rod; 𝑢,𝑣,𝜔 - displacement projections on the orts small angles of rotation of the Frenet trihedron during oscillations around the unit vectors projections of changes in the main vector of curvature of the elastic line during oscillations. In addition, here it is indicated: A and B - bending rigidity of the rod: C - torsional rigidity: 𝑆− cross-sectional area; 𝜌− density; 𝑙- is the length of the elastic line.
System of equations (3) is reduced to one equation for the unknown 𝜔:

(7)
t-time. (8)
In equation (7) we move on to the variable φ varying from zero to , according to formula (2), where
(9)
The main assumption is that φ₀ is a large number and q_0 (φ) changes slowly. Let us introduce a small parameter
(10)
and a new variable
(11)
Then
(12)
Moreover, the quantities are of the order of unity compared to μ. Let's denote
(13)
To determine the natural frequencies ω_k and the eigenfunction ω_k (φ) we put in equation (7)
)
Then (omitting the index ) we get:

(15)
Where
(16)
System of equations (5) is reduced to two equations for the unknowns and . Believing
=
(17)
To determine the eigenfunctions and , we obtain a system of equations:



(18)
(19)
Where
The general solution to the problem of forced oscillations in the plane of a curve can be represented as a series of eigenfunctions
(20)
Where
(21)
Likewise,( Similarly)
(22)
Where
Here , mean the projections onto the vectors of the external force per unit length of the rod.