## Abstract

The transverse vibrations of an axially moving string that is subjected to a distributed follower force are examined here. This model provides an insight into the complex dynamics of seemingly simpler systems such as silicon wafer cutting using wire saws, and aerial or marine towing, where a relatively long flexible structure is dragged through fluid. The equation of motion is derived and it includes the axial variation in the tension that arises due to acceleration and the follower force. As the exact analytical solution of this equation is difficult to determine, the approximate closed-form modal solution of a non-travelling counterpart of the system is obtained using the asymptotic technique, which is then used as a basis to obtain the numerical solution for the axially moving string. The effect of the follower force and viscous dissipation on the eigenstructure of the system is investigated. Mathematical operations such as the Hermite form and the Routh–Hurwitz criterion are applied to the characteristic polynomial to investigate the dynamic behaviour of these modes. The semi-analytical approach presented explains the ‘mathematical’ instability (in the absence of damping) that arises when both axial transport and follower force are simultaneously present. An unusual transition of the dynamic behaviour from the stable to the overdamped and then directly to the unstable regime is observed.

## 1. Introduction

Axially moving continuous structures occur in several engineering and technological systems such as in chain and belt drives, magnetic tape, elevator cables, band saws and high-speed fibre winding. A basic characteristic which distinguishes these systems from their corresponding non-travelling counterparts is their velocity-dependent natural frequencies. Also, as such continuous structures vibrate in the transverse direction, which is superposed on a rigid body translation in the axial direction, an element on the system travels in a curved path with respect to a fixed frame of reference. This motion gives rise to terms corresponding to the centripetal and Coriolis (gyroscopic) components of acceleration in the equation of motion. The presence of these acceleration terms renders the dynamics of such systems complex. The transverse vibrations of such systems have non-constant spatial phase and any disturbances in the system travel at different speeds in the upstream and downstream directions. Moreover, the total mechanical energy of such systems is non-constant, and they are hence non-conservative.

The equation of motion of such axially moving systems is also equivalent to that of pipes conveying fluid; there, the gyroscopic effect occurs due to the axial motion of fluid along with the transverse vibration of the pipe [1]. The stability of such pipes conveying fluid can be qualitatively compared with that of a column subjected to a non-conservative tangential follower force at the end, but with the additional presence of a gyroscopic effect [2]. Thus, the dynamic analysis of strings/beams subjected to follower-type loads can help in better understanding the rich dynamical behaviour of seemingly simple systems such as pipes conveying fluid. Wang [3] recently investigated the possibility of flutter instability in pipes conveying fluid which are subjected to such a distributed follower force. Recently, Stangl provided an alternative approach combining extended Lagrange equations and the Ritz method to derive the nonlinear equation of motion considering large vibrations of pipes conveying fluid [4]. Ibrahim [5] presented a recent review on pipes conveying fluid including the nonlinear models of such systems.

Another mathematically equivalent, but physically different, example of the pipe conveying fluid is the beam travelling axially through a fluid. The stability of such beams subjected to fluid interaction has been investigated earlier [6–8]. When such a beam vibrates in the direction transverse to its motion, it experiences a drag force which is instantaneously inclined to the direction of axial motion. This force can be resolved into two components, one parallel to the instantaneous slope of the beam, and the other, perpendicular to it. In the limiting case, when the diameter/width of the beam is negligible, it can be treated as a string. In this case, the parallel component can be related to a distributed follower force while the perpendicular component (for small deflections) acts as damping on the transverse motion. Thus, to better understand the dynamics, in this paper, a simpler system consisting of an axially moving string subjected to a distributed follower force in the presence of viscous damping is considered.^{1} (For better understanding of the equivalence between parameters used in the presented model and the fluid forces experienced by a string, see appendix A, electronic supplementary material.) A detailed analysis of the effect of this distributed follower force on the dynamics of the system is presented. This force is always oriented in a direction tangential to the instantaneous slope of the string at every location. Such a model is prototypical of cases where a long, flexible structure is dragged through fluid, as found in silicon wafer cutting processes. The instability of the wire can affect the surface quality and material loss (termed as kerf loss) during the operation of silicon wafer slicing, and hence affect the overall productivity of the operation (which has significant implications for the prices of, for instance, solar photo voltaic cells that are made using these wafers).

Early research in the area of axially moving systems was limited to strings moving with constant axial speed [10–12]. Wickert & Mote [13] extended the model and analysed the transverse vibrations of axially moving strings and beams under arbitrary excitation and initial conditions using modal analysis and Green’s function method. Tan & Ying [14] presented the solution in frequency domain for the response of an axially moving string with general boundary conditions. Parker [15] examined the supercritical stability of an axially moving string supported by an elastic foundation. Apart from this research on linear dynamics, there is some work which includes nonlinearity by considering viscoelastic effects and large deformations. Yang and colleagues [16,17] presented asymptotic analysis and approximate solutions of axially moving viscoelastic strings and beams. Very recently, Vetyukov [18] proposed a nonlinear finite element model of large vibrations of axially moving strings and beams using a mixed Eulerian–Lagrangian description. Miranker [19] was the first to derive the equation of motion for the transverse vibration of an accelerating string. This equation has been used by many researchers for further analysis where the axial speed of the system is considered to have periodic fluctuations [20–25]. However, Miranker’s equation does not correctly describe the dynamics of an axially accelerating string because the effect of axial variation in the tension arising due to acceleration was not accounted for in the equation. Yang *et al.* [26] have also investigated parametric resonances in viscoelastic accelerating strings using an asymptotic approach.

In this work, the equation of motion is developed for an axially accelerating string considering this variation in the tension when it is subjected to an axially distributed follower force. (The equation itself can, alternatively, also be obtained by reducing those of certain other mathematically equivalent systems, such as that of pipes conveying fluid, as in [3].) The effect of follower force is analysed for the case in which the string is moving with constant axial speed. First, the approximate closed-form solution is obtained using asymptotic techniques, when the distributed follower force is acting on the non-travelling string. The mode shapes obtained using the approximate analytical method are used as basis functions to obtain the numerical solution for an axially moving string subjected to the follower force. The approximate closed-form solution is used to directly obtain the eigenstructure and stability regimes, and understand their dependence, separately, on the (non-conservative) follower force and the (intutively stabilizing) viscous dissipation. The effect of this force on the critical speed of the moving string is examined and critical curves distinguishing the different dynamic regimes are obtained.

## 2. Physical model

Figure 1 schematically shows a planar model of an inextensible string moving axially towards the right with a time-varying axial speed ‘*v*(*t*)’. The string is subjected to an axially distributed constant follower force per unit length ‘*p*’. The key characteristic of the follower force is that, it is, at every location, always oriented parallel to the instantaneous slope of the string at that location, as is found in classic Beck [27] and Pflüger columns [28]. Thus, the follower force is different from a simple axial force which is observed, for example, as the weight due to gravity in the case of a vertical hanging string. Apart from an axial component, this follower force has a transverse component that depends upon the instantaneous slope of the segment over which the force acts. The tension at the trailing (left) end of the string is ‘*T*_{0}’, which increases axially due to the presence of the follower force as one moves towards the winding (right) end. The string is also subjected to a distributed viscous damping force with damping coefficient ‘*c*’. Note that any Coulomb friction that arises between the string and the surface with which it is in contact, such as friction guides or the workpiece in a wire-sawing process, can be approximated as viscous damping when the travelling speed of the string is much higher than the maximum speed of its transverse oscillations [29]. The displacement of both ends of the string is assumed to be restricted in the transverse direction at the pulleys, which are separated by the span of length ‘*l*’. The instantaneous vertical displacement of each segment of the string is denoted as ‘*w*(*x*,*t*)’.

In much of the prior literature on the transverse vibrations of axially accelerating strings, the equation of motion derived by Miranker in the 1960s [19] is used. This equation does not consider the variation in axial tension that arises due to the acceleration itself. However, the balance of forces in the horizontal direction for any segment of the string at location ‘*x*’ (shown in figure 1) gives the variation in the tension to be
*ρ*’ is the mass per unit length of the string. In equation (2.1), the terms ‘*ρx* *dv*/*dt*’ and ‘*px*’ represent the effect of the presence of the time-varying axial speed and the follower force, respectively. The tension ‘*T*_{0}’ at the trailing end of the string increases axially with ‘*x*’ due to these two terms. (Note that Miranker’s model considers *T*(*x*)=*T*_{0}, i.e. uniform tension along the length.)

When the string is displaced from equilibrium in the transverse direction as it undergoes axial rigid body translation, the transverse velocity of each segment of the string depends also upon its horizontal travelling speed. This transverse velocity is given by *V* _{tr}=*v*(∂*w*/∂*x*)+∂*w*/∂*t*. As the viscous damping in this case is proportional to the transverse velocity, it can be written as^{2}
*v*(•)_{,x}+(•)_{,t}, the balance of forces in the vertical direction can be rewritten as

The terms ‘*ρv*^{2}*w*_{,xx}’ and ‘2*ρvw*_{,xt}’ correspond to the centripetal and Coriolis components of acceleration, respectively, that appear due to the simultaneous transverse displacement and axial rigid body translation. Substituting equation (2.1) into equation (2.3), the equation of motion of transverse vibration is obtained as
*c*_{0}’ represents the wave speed ‘^{3} (Also, in [3], the tension at the winding end is prescribed, whereas in this work, the tension at the trailing end is prescribed. Of course, in both the cases, the tension at the winding end is always greater than that at the trailing end.) As the transverse displacement of both the ends is restricted, the boundary conditions become *w*(0,*t*)=*w*(*l*,*t*)=0. Using the non-dimensional terms

## 3. Insights from the non-travelling case: v ¯ = 0

To understand the effect of the follower force on the dynamics, first a non-travelling string (with no axial speed) but subjected to a follower force is considered. This can be considered to be equivalent to the physical case in which a string is placed in the axial flow of fluid with both its ends fixed. The approximate closed-form solution presented for such a case in this section will assist in obtaining faster numerical convergence for the original problem of interest. Also, this mathematical exercise can provide better insight into the effect of follower force and damping on the variation of the eigenstructure of the actual problem of the axially moving string. The equation of motion of the undamped transverse vibration of a string in the absence of axial speed subjected to an axially distributed follower force acting towards the left is obtained by substituting ** η**(

*t*) is a vector of size (

*N*×1), and

**,**

*M***and**

*K***are matrices of size (**

*C**N*×

*N*). This equation is written in the state-space form

**and**

*C***will vary according to the particular equation.) The eigen-value problem is obtained, the eigen-values of which represent the natural frequencies and modal damping of the system, and whose eigen-vectors correspond to the mode shapes. The nature of the eigen-values can be used to determine the stability of the system.**

*K*The variation of the imaginary part of the first six eigen-values of the discretized system is shown in figure 2*a* when only the transverse component of the follower force is present (equation (3.3)), and in figure 2*b* with only the axial component being present (equation (3.2)). Interestingly, figure 2*a* shows that a flutter-like instability occurs when the transverse component of the follower force increases. To verify this, the mode shapes of the system with only the transverse component being present, and with follower force magnitudes marked by dashed lines in figure 2*a* are plotted in figure 3. At the onset of instability (between *b*) increase monotonically with the follower force; this variation dominates over that of the transverse component when the entire follower force is present. Note that this is always true for the case in which the small angle assumption holds true, because the transverse component at any location can never be greater than the axial component in such a case. Therefore, the natural frequency of a string that is subjected to a follower force increases with an increase in the force’s magnitude. (The variation of natural frequencies of the string subjected to the complete follower force is shown later in figure 6). It is clear from equations (3.2) and (3.3) that the transverse and axial components negate the effect of each other to some extent because the signs of the ‘

In figure 4*a*, during the initial time instants, the right half-segment of the string is pushed upwards, whereas the left half is pulled downwards due to the corresponding nature of the transverse components, whereas if only the axial component of the follower force is considered, the tension varies axially and is higher in the segment at the winding (right) end than that at the trailing (left) end. Thus, in figure 4*b*, the string displaces less in the transverse direction as one moves towards the right end. However, when both the components are simultaneously considered, the effect of axial variation in the tension dominates over that of the transverse component; and hence, the overall time response is skewed towards the left in the presence of the follower force, as shown in figure 4*c*. Even though the axial and transverse components of follower force cannot be separated in reality, this analysis enables one to gain more insight into the effect of follower force on mode shapes and the nature of the time response.

### (a) Approximate analytical solution

Having examined the effect of the individual components, in this section, the approximate closed-form solution is presented for the non-travelling string subjected to the complete follower force and in the presence of viscous damping. Note that due to the presence of the spatially varying coefficients of a differential term, the exact analytical solution of the equation can be expressed in terms of Bessel functions; however, closed-form expressions for the natural frequencies and mode shapes cannot be obtained using that approach. Lubbad [30] presented approximate closed-form solutions using asymptotic techniques for the vibrations of vertical strings and beams subjected to gravity. A similar approach is now used to find the solution in this case. Though this method provides an approximate solution, closed-form expressions of the natural frequencies and mode shapes can be obtained, which provide better insight into the effect of certain variables (follower force and damping coefficient in this case) on the variation of eigenstructure. The asymptotic technique is a local analysis approach which uses a power series solution, in which the expansion series is divergent. In this type of local analysis, the nature of the differential equation is determined from the singular points, i.e. the points which are not analytic.^{4} To solve equation (2.6) with zero axial speed, the transverse displacement of any segment is assumed to be of the variable separable form *z*=1/*t*. Thus, *du*/*dz*=(*du*/*dt*)(*dt*/*dz*)=−*t*^{2}(*du*/*dt*) and *d*^{2}*u*/*dz*^{2}=*t*^{4}(*d*^{2}*u*/*dt*^{2})+2*t*^{3}(*du*/*dt*). Substituting back into equation (3.6) and simplifying, we get
*t*=0, and thus, equation (3.6) has an irregular singular point at *z*. Assuming *u*=*e*^{s(z)}, equation (3.6) can be rewritten as
*s*′′(*z*)≪[*s*′(*z*)]^{2}, neglecting *s*′′(*z*), the above equation is transformed into a quadratic equation in *s*′(*z*). The roots of this equation are *u*(*x*) can be written as *u*(0)=*u*(1)=0, we get
*c*_{1} and *c*_{2} should vanish, which gives
_{n}

The approximate closed-form expression for the mode shapes of the undamped system can be written as

The natural frequencies obtained by using the approximate closed-form equation (3.17) are compared in figure 6 with those obtained by using the numerical method of solution detailed earlier in §3. It is seen that the natural frequencies obtained using both the methods match closely. The deviation, however, increases as the magnitude of the follower force increases, which is expected because the closed-form expression is valid only for large values of *z* starts to decrease and the solution deviates from the actual value. The accuracy can, however, be increased by considering the constant of integration which was neglected earlier to find *s*(*z*). Also, note that the natural frequencies of the system increase as the follower force increases, since the axial component of the force dominates as discussed earlier, which results in an increase in the average tension. The mode shapes obtained using equation (3.18) are plotted in figure 7 for different magnitudes of the follower force. These mode shapes are compared with those of the non-travelling string without the follower force being present. When the follower force is present and acts towards the left, the axial component dominates and the tension increases axially towards the winding (right) end. Thus, a segment of the string towards the right displaces less in the transverse direction than one towards the left; hence, the mode shapes skew towards the left as the follower force increases. Note that these mode shapes obtained using the approximate analytical approach also satisfy the boundary conditions of the original system of an axially moving string subjected to a distributed follower force, and can now be used as the basis functions in the expansion series in the numerical solution. This leads to faster convergence, with a fewer number of terms in the approximation.

## 4. The case of the moving string

To analyse the effect of the follower force on the moving string, equation (2.6) is discretized into the form of equation (3.4), where ** C**=

**+**

*G*

*C*_{1}and

**=**

*K*

*K*_{1}+

*K*_{2}, and the elements of these matrices are written as

*T*

_{0}.)

Figure 9 shows the first three mode shapes of the string when it is moving axially with different speeds and is subjected to different magnitudes of the distributed follower force. These mode shapes are obtained using the eigen-vectors from the eigen-value problem defined by equation (3.4) without the presence of viscous damping, and are compared with those of the non-travelling string without the follower force. The first row shows the mode shapes of the non-travelling string and the dashed lines represent the modes when no follower force is present. Thus, the dashed lines in the first row correspond to the mode shapes of a simple non-travelling string which can be mathematically represented using sinusoids. As the string is moving towards the right, as the axial travelling speed increases, the modes are skewed towards the left due to the effect of inertia and the gyroscopic terms. When the follower force is present, the mode shapes are skewed further towards the left as the follower force increases, as discussed earlier in §3a.

### (a) Variation of critical speed

It is well known that the natural frequencies of a moving string decrease as the travelling speed increases, and finally vanish at a certain speed termed as the critical speed. As the presence of the follower force changes the natural frequencies of the system, a change in the critical speed is also expected. Figure 10 shows the variation of the first natural frequency with axial speed as the follower force is varied. The critical speed increases as the follower force increases, similar to the increase in the natural frequency in the presence of the follower force as was seen in figure 8. The variation of this critical speed with the follower force corresponding to the first natural mode of vibration is shown by the red curve in figure 11. The region to the right of this curve represents those parameter combinations for which the first natural frequency vanishes and the first natural mode of the system becomes unstable. Thus, this curve represents the transition between stability and instability (and vice versa) of the first natural mode corresponding to different magnitudes of follower force. For example, if a follower force of 10 units were to act on the string, the string would be constrained to move with an axial speed below 2.45 units in order for the first mode to be stable; alternatively, if the string were to move with an axial speed of 2.45 units, a follower force with a magnitude of at least 10 units should act on the string in order for the first mode to be stable. Note that when no follower force is present, the curve passes through

This analysis is performed by solving the discretized equation of motion discussed earlier using a one-mode approximation in the expansion. The matrix ** C** in equation (3.4) is, incidentally, a null matrix in the one-mode approximation (i.e. the effect of the gyroscopic term does not appear in this approximation). In the absence of damping, the matrices

**and**

*M***, and hence**

*K***in the state-space form**

*A*^{2}+

*a*

_{0}=0, where λ represents an eigen-value and

*a*

_{0}>0, both the roots of the characteristic polynomials (or eigen-values) are purely imaginary and represent the natural frequencies of the system. When

*a*

_{0}<0, the eigen-values are purely real and have no imaginary part, i.e. the natural frequency vanishes. When

*a*

_{0}=0, the eigen-values become exactly zero, which is shown by the critical curve in figure 11. The closed-form expression for this critical curve is

*a*

_{0}<0. While, this analysis is performed using only a one-term approximation, the solution obtained by using the two-term approximation differs greatly, especially after the critical speed is crossed. Also, all the solutions obtained using the even-term approximation have common properties which differ from those obtained using odd-term approximations. Pakdemirli [20] suggested that even-term approximations yield better results because the gyroscopically coupled terms arise only in those cases.

### (b) Critical curves and transition between dynamic regimes

The variation of natural frequencies with axial speed is shown in figure 12 for the moving string, with and without the follower force. When no follower force is present, it is clear that, as the speed increases, the natural frequencies of the string start decreasing and all of them vanish at a single speed

Using a similar approach as used in the case of the one-mode approximation, closed-form expressions for the critical curves can be obtained. For an ‘*n*’-mode approximation, the characteristic equation |*A*−*λI*|=0 is written as the polynomial
*a*_{0},*a*_{1},…,*a*_{2n−1} are functions of

In equation (4.1), at least one eigen-value will be zero if *a*_{0}=0. Thus, the closed-form expression relating the parameters *k*_{0} is a constant. In this case, *a*_{0}=0 has only one valid repeated solution, i.e. *a*_{0}=0 results in a polynomial of order 2*n* in the variable

Figure 13*b* shows such dynamic regimes when the analysis is performed using a two-mode approximation. The hatched regions (blue as well as black) correspond to the regime where the imaginary part of at least one eigen-value vanishes. The corresponding variation of the signature of the Hermite matrix with axial speed is shown in figure 13*a* when *b*. The critical curves obtained using the approach explained earlier (i.e. the curves corresponding to ‘*a*_{0}=0’ with a two-term approximation), when superimposed onto the hatched region, distinguish the overdamped and unstable regions corresponding to that mode. Note that when no follower force is present, both the curves merge together at *b*. Thus, as the speed increases, the real part attains a positive value upon crossing this curve and the corresponding mode becomes unstable. The real branch further starts decreasing and goes from positive to negative, crossing another critical curve, and that mode then enters an overdamped regime. When the axial speed is further increased, the imaginary part again becomes non-zero and the two branches of the real part merge together, so that the eigen-values are again complex conjugates of each other.

### (c) ‘Mathematical’ instability

In the absence of viscous dissipation, the matrix ** C_{1}** vanishes and

**=**

*C***, which is a skew-symmetric matrix. As the diagonal elements of the skew-symmetric matrix are zero, it can be seen that all the diagonal elements of the state matrix**

*G***become zero. As the sum of all the eigen-values of a matrix is equal to its trace, we can write**

*A**n*is the number of terms in the expansion series in equation (3.4) and λ

_{i}’s are the eigen-values. This results in two sets of possibilities: either all the eigen-values are purely imaginary and complex conjugates of each other (i.e. the system is marginally stable), or the real parts of some of the eigen-values are positive (i.e. the system is unstable). Note that in the first case where all the eigen-values are purely imaginary, the characteristic polynomial can be written as

*β*

_{i}’s are real constants, which product contains only even powers of λ. This suggests that, in this case, for a system to be marginally stable, the terms corresponding to the odd powers of λ in the characteristic polynomial should vanish. However, in the simultaneous presence of both a follower force and axial motion such odd powers of λ can appear in the characteristic polynomial; this implies the second possibility where the real parts of some of the eigen-values are positive, thereby rendering the system unstable. Figure 14 shows the variation of the imaginary part of the eigen-values with the number of terms in the approximation. As the number of terms in the expansion series in equation (3.4) is increased, the eigen-values converge to a specific value; however, some of the eigen-values corresponding to the higher modes are always unstable. As the number of terms increases, the unstable eigen-values of the lesser term approximations become stable; however, a few other higher eigen-values remain unstable. The presence of even one unstable eigen-value can possibly lead to instability of the entire system. Such ‘mathematical’ instability of the higher modes is a characteristic of the governing equation (2.6). To illustrate further, the Routh–Hurwitz criterion [32] is now used to understand the nature of the characteristic polynomial obtained while solving the eigen-value problem. According to this criterion, the necessary condition for the polynomial to be asymptotically stable is that the coefficients of all the terms in the polynomial have to be present and be positive. Note that the coefficient of the term that has the second highest power (

*a*

_{2n−1}) in the characteristic equation (4.1) represents the summation of eigen-values, which is zero in the absence of viscous damping. Hence, mathematically, this system can only be marginally stable and not asymptotically stable.

To illustrate this, the characteristic polynomial for a two-mode approximation of the undamped system is λ^{4}+*a*_{2}λ^{2}+*a*_{1}λ+*a*_{0}=0. According to the Routh–Hurwitz criterion, the fourth-order polynomial of the form *a*_{4}*s*^{4}+*a*_{3}*s*^{3}+*a*_{2}*s*^{2}+*a*_{1}*s*+*a*_{0}=0 has stable roots if the coefficients satisfy the relations
*a*_{3}=0, the system can at most be marginally stable if *a*_{1} is also zero. For a two-mode approximation, *a*_{1} is obtained to be

However, in reality, the string is always subjected to some damping. When viscous damping is considered in equation (3.4), the matrix ** C** contains terms due to both

**and**

*G***, and hence the diagonal elements of**

*C*_{1}**do not vanish. In such cases, the term**

*M*^{−1}*C**a*

_{2n−1}in the characteristic equation (which represents the trace of the state matrix

**) becomes non-zero and the system can be asymptotically stable for certain non-trivial combinations of**

*A*Figure 15*a* shows the root locus plot, i.e. the variation of the imaginary and real parts of the eigen-values, for an axially moving string subjected to a follower force (*a*_{2n−1} term. A similar variation is shown in figure 15*b* in the presence of damping (

### (d) Stability regimes in the presence of viscous damping

Figure 17*a* shows the critical curves in the presence of different values of damping which are obtained using a procedure similar to that discussed earlier in §4b (i.e. by obtaining the roots of the equation ‘*a*_{0}=0’) using a two-term approximation. As explained earlier, when no damping is present, the critical curves merge together at *b*. As the damping coefficient increases, the real part of the eigen-value becomes more negative and hence the curves in figure 17*b* shift downwards. Thus, for the same magnitude of the follower force, the real part becomes positive over a smaller range of axial speeds in the presence of damping than for the case when damping is absent. This range, over which the natural mode enters the unstable region, reduces further as the damping coefficient increases.

Figure 18*a* shows the critical curves in the presence of damping (*b* for different magnitudes of the follower force. The elliptical branch of this variation corresponds to the velocity regime in which the imaginary part of the eigen-value vanishes. Clearly, as the magnitude of the follower force increases, the elliptical branch enlarges and shifts towards the right. In the presence of damping, when the follower force has magnitude which is less than a certain value (*a*_{0}=0’ happen to be complex; this implies that the eigen-value cannot be zero for any real axial speed. In such cases, the elliptical branch of the variation of the real part is so small that it does not change its sign from negative to positive for any regime of axial speed. As the follower force increases and reaches a particular value (*a*_{0}=0’ has repeated real roots and the corresponding elliptical branch just touches zero at that value of speed. When the follower force is increased further, this branch has a positive value for some regime of axial speed whose limits can be obtained by finding the real roots of the equation ‘*a*_{0}=0’. The equation in this case has two distinct roots for *a*_{0}=0’.

## 5. Summary

The dynamics of an axially moving string subjected to a distributed follower force have been investigated here. The tension varies axially in the string due to both acceleration and the follower force. The equation of motion is derived considering this variation and is found to be different from the conventionally used equation due to Miranker. The follower force acts tangentially to the instantaneous slope of the string, and is, thus, different from a simple axial force such as gravity acting on a vertical string. The axial component of the follower force dominates over the transverse one for the cases in which the small angle assumption holds. Closed-form expressions for the natural frequencies and the mode shapes are obtained for a non-travelling string subjected to the follower force using asymptotic techniques. Further, the transverse vibrations are examined when the string moves with a constant axial speed in the presence of the follower force. The natural frequencies and, hence, the critical speeds increase with the follower force due to an increase in the average tension in the string. All the natural frequencies of the moving string do not vanish at the same speed in the presence of the follower force. Individual modes exhibit different dynamic behaviour in certain speed regimes. An unexpected transition from stable to overdamped and then to unstable dynamic behaviour is seen, and critical curves separating such overdamped and unstable regimes are obtained. The follower force sometimes has a stabilizing effect (for example it increases the critical speed at which the natural frequency vanishes); however, it sometimes can lead to instability. To determine the reasons for such mathematical instability, the Routh–Hurwitz criterion is applied, and in the absence of damping, the system is found to be stable only if either the axial speed or follower force or both vanish. In the presence of sufficient viscous damping, the system becomes stable and until some critical value of the follower force, individual modes do not transition between the overdamped and unstable regimes. Once the follower force exceeds this value, such transitions occur. The mathematical techniques used in this work and the presented dynamic analysis can aid in better understanding of the dynamics of physical systems such as pipes conveying fluid, aerial or marine towing, textile machinery and silicon wafer cutting where long flexible structures are dragged through fluid. Further, the presented analysis for the fundamental element of a string can be extended to two-dimensional axially moving elements (such as beams and plates), which find applications in the material processing industry, where long thin sheets of metal or polymer travel between supports and are subjected to fluid interaction during processing.

## Data accessibility

This paper contains no experimental data.

## Authors' contributions

Both the authors contributed to the formulation of the mathematical model and the drafting of the manuscript. V.K. planned the overall study. P.L. developed the fundamental concepts and performed the detailed mathematical analyses and the numerical simulations.

## Competing interests

We declare that we have no competing interests.

## Funding

No funding has been received for this article.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4079342.

↵1 When an inclined cylinder is subjected to a fluid flow, the drag normal to the surface is typically divided into components arising due to pressure and skin friction. When the diameter of the cylinder is very small, the component due to pressure becomes negligible and the component only due to skin friction is significant [9]. When this normal force is included in the equation of motion, it creates an effect similar to that of viscous damping, whereas the drag force tangential to the surface acts as a follower force.

↵2 The notations denote (•)

_{,x}=(∂/∂*x*)(•), (•)_{,xt}=(∂^{2}/∂*x*∂*t*)(•), (•)_{,xx}=(∂^{2}/∂*x*^{2})(•) and (•)′=(*d*/*dt*)(•)↵3 It is safe to neglect these terms, especially in applications such as wire sawing, where metallic wires with very small diameters are used.

↵4 ‘Analytic functions’ are functions which are infinitely differentiable and whose Taylor’s series is convergent. If the function is not analytic at some points, then those points are called ‘singular points’. To understand the nature of the differential equation, consider the second-order differential equation

*y*′′+*a*(*x*)*y*′+*b*(*x*)=0, where the primes denote the derivatives with respect to*x*. If*a*(*x*) and*b*(*x*) are analytic near the point*x*_{0}, then it is termed as an ‘ordinary point’, otherwise it is termed as singular. Further, if (*x*−*x*_{0})*a*(*x*) and (*x*−*x*_{0})^{2}*b*(*x*) are analytic near*x*_{0}, then it is termed a ‘regular singular point’. If*x*_{0}is neither ordinary nor regular, it is called an ‘irregular singular point’. The solutions for the differential equations with ordinary points can be obtained by using the Taylor’s series expansion, whereas those with regular singular points can be solved using the Frobenius method. However, both these methods have convergent expansion of the series. If the differential equation contains irregular singular points, the solution cannot be obtained using these methods and asymptotic methods can be used [31].

- Received November 4, 2017.
- Accepted April 10, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.