Nonlinear planar modeling of massive taut strings travelled by a force-driven point-mass

Abstract

The planar response of horizontal massive taut strings, travelled by a heavy point-mass, either driven by an assigned force, or moving with an assigned law, is studied. A kinematically exact model is derived for the free boundary problem via a variational approach, accounting for the singularity in the slope of the deflected string. Reactive forces exchanged between the point-mass and the string are taken into account via Lagrange multipliers. The exact model is consistently simplified via asymptotic analysis, which leads to condense the horizontal displacement as a passive variable. The dynamic increment of tension, with respect the static one, is neglected in the governing equations, but evaluated a posteriori, as a higher-order quantity in a perturbation perspective. The equations are solved and rearranged in the form of an integral equation coupled with an integro-differential equation, thus extending a procedure already introduced in the literature. Numerical results, showing the importance of the horizontal reactive force on the quality of motion, are discussed, generalizing those relevant to massless strings.

Appendices

Appendix A: Variational derivation of the equations of motion

The Equations of motion (14)–(18) are derived by the stationary condition of the modified Hamilton principle (9), $\delta \tilde{{\mathcal {H}}}\left[ {\mathbf {r}},{\mathbf {x}},\xi ,{\mathbf {R}}\right] =0$, i.e.,

$$\begin{aligned} \begin{aligned}&\intop _{t_{1}}^{t_{2}}\left[ \delta {\mathcal {K}}_{m}+\delta {\mathcal {K}}_{s}-\delta {\mathcal {U}}_{s}+\delta {\mathcal {W}}-{\mathbf {R}}\cdot \left( \delta {\mathbf {x}}-\delta \hat{{\mathbf {r}}}\right) \right] \,\mathrm {d}t\\&\quad -\intop _{t_{1}}^{t_{2}}\delta {\mathbf {R}}\cdot \left( {\mathbf {x}}-{\mathbf {r}}\left( \xi \left( t\right) ,t\right) \right) \,\mathrm {d}t=0\qquad \forall \left( \delta {\mathbf {r}},\delta {\mathbf {x}},\delta \xi ,\delta {\mathbf {R}}\right) , \end{aligned} \end{aligned}$$

(71)

where the definitions (10) hold for ${\mathcal {K}}_{m},{\mathcal {K}}_{s},{\mathcal {U}}_{s},{\mathcal {W}}$ and (13) for $\delta \hat{{\mathbf {r}}}$.

The variations of non-integral in space terms are straightforward, namely:

$$\begin{aligned}&\intop _{t_{1}}^{t_{2}}\delta {\mathcal {K}}_{m}dt =\intop _{t_{1}}^{t_{2}}M\,\dot{{\mathbf {x}}}\cdot \delta \dot{{\mathbf {x}}}\,\mathrm {d}t=-\intop _{t_{1}}^{t_{2}}M\,\ddot{{\mathbf {x}}}\cdot \delta {\mathbf {x}}\,\mathrm {d}t, \end{aligned}$$

(72)

$$\begin{aligned}&\delta {\mathcal {W}} ={\mathbf {F}}\cdot \delta {\mathbf {x}}, \end{aligned}$$

(73)

where terms evaluated at $t_{1},t_{2}$ after integration by parts have been canceled. The variation of the terms which involve integrals on space, instead, is more complicated, since it calls for properly accounting for the presence of a singularity at $s=\xi$. By breaking the integration interval as:

$$\begin{aligned} \intop _{0}^{\ell }f\left( s\right) \,\mathrm {d}s=\intop _{0}^{\xi }f\,\mathrm {d}s+\intop _{\xi }^{\ell }f\,\mathrm {d}s, \end{aligned}$$

(74)

it follows that:

$$\begin{aligned} \delta \intop _{0}^{\ell }f\left( s\right) \,\mathrm {d}s= & {} \intop _{0}^{\ell }\delta f\,\mathrm {d}s+\left( \intop _{0}^{\xi +\delta \xi }f\,\mathrm {d}s-\intop _{0}^{\xi }f\,\mathrm {d}s\right) \nonumber \\&+\left( \intop _{\xi +\delta \xi }^{\ell }f\,\mathrm {d}s-\intop _{\xi }^{\ell }f\,\mathrm {d}s\right) \nonumber \\= & {} \intop _{0}^{\ell }\delta f\,\mathrm {d}s+\left[ f\left( \xi _{-}\right) -f\left( \xi _{+}\right) \right] \delta \xi \nonumber \\= & {} \intop _{0}^{\ell }\delta f\,\mathrm {d}s-\llbracket f\rrbracket \delta \xi , \end{aligned}$$

(75)

Accordingly:

$$\begin{aligned}&\delta {\mathcal {K}}_{s}=\intop _{0}^{\ell }m\,\dot{{\mathbf {r}}}\cdot \delta \dot{{\mathbf {r}}}\,\mathrm {d}s-\frac{1}{2}m\,\llbracket \dot{{\mathbf {r}}}\cdot \dot{{\mathbf {r}}}\rrbracket \delta \xi , \end{aligned}$$

(76)

$$\begin{aligned}&\delta {\mathcal {U}}_{s}:=\intop _{0}^{\ell }\frac{T}{1+\varepsilon }\,{\mathbf {r}}'\cdot \delta {\mathbf {r}}'\,\mathrm {d}s-\llbracket \phi \rrbracket \delta \xi , \end{aligned}$$

(77)

in which $\delta \phi =\frac{\partial \phi }{\partial \varepsilon }\,\delta \varepsilon =T\,\delta \varepsilon$ has been exploited, together with $\delta \varepsilon =\frac{{\mathbf {r}}'\cdot \delta {\mathbf {r}}'}{1+\varepsilon }$, following Eq. (2).

Next step calls for integrating by parts the last two expressions. By noticing that:

$$\begin{aligned} \begin{aligned}&\intop _{0}^{\ell }f\left( s,t\right) {\dot{g}}\left( s,t\right) \,\mathrm {d}s\\&\quad = -\intop _{0}^{\ell }{\dot{f}}\,g\,\mathrm {d}s+\intop _{0}^{\ell }\left( f\,g\right) ^{\bullet }\,\mathrm {d}s\\&\quad = -\intop _{0}^{\ell }{\dot{f}}\,g\,\mathrm {d}s+\frac{\mathrm {d}}{\mathrm {d}t}\left( \intop _{0}^{\ell }f\,g\,\mathrm {d}s\right) +{\dot{\xi }}\llbracket f\,g\rrbracket ,\\&\intop _{0}^{\ell }f\left( s\right) g'\left( s\right) \,\mathrm {d}s\\&\quad = -\intop _{0}^{\ell }f'g\,\mathrm {d}s+\left[ f_{-}g_{-}\right] _{0}^{\xi }+\left[ f_{+}g_{+}\right] _{\xi }^{\ell }\\&\quad = -\intop _{0}^{\ell }f'g\,\mathrm {d}s+\left[ f_{\,}g\right] _{0}^{\ell }-\llbracket f\,g\rrbracket , \end{aligned} \end{aligned}$$

(78)

it follows:

$$\begin{aligned} \intop _{t_{1}}^{t_{2}}\delta {\mathcal {K}}_{s}\,\mathrm {d}t=&-\intop _{t_{1}}^{t_{2}}\intop _{0}^{\ell }m\,\ddot{{\mathbf {r}}}\cdot \delta {{\mathbf {r}}}\,\mathrm {d}s\,\mathrm {d}t+\intop _{t_{1}}^{t_{2}}m\,{\dot{\xi }}\llbracket \dot{{\mathbf {r}}}\cdot \delta {\mathbf {r}}\rrbracket \,\mathrm {d}t \end{aligned}$$

(79)

$$\begin{aligned}&-\intop _{t_{1}}^{t_{2}}\frac{1}{2}m\llbracket \dot{{\mathbf {r}}}\cdot \dot{{\mathbf {r}}}\rrbracket \delta \xi \,\mathrm {d}t, \end{aligned}$$

(80)

(81)

where the arbitrariness of $t_{1},t_{2}$ and the external boundary conditions were accounted.

The last step concerns the treatment of the discontinuities at $s=\xi$. By remembering Eq. (13b), it follows:

$$\begin{aligned} \begin{aligned} \llbracket {\mathbf {f}}\left( s\right) \cdot \delta {\mathbf {r}}\rrbracket&={\mathbf {f}}_{+}\cdot \delta {\mathbf {r}}_{+}-{\mathbf {f}}_{-}\cdot \delta {\mathbf {r}}_{-}\\&={\mathbf {f}}_{+}\cdot \left( \delta \hat{{\mathbf {r}}}-{\mathbf {r}}'_{+}\delta \xi \right) -{\mathbf {f}}_{-}\cdot \left( \delta \hat{{\mathbf {r}}}-\delta \xi \,{\mathbf {r}}'_{-}\right) \\&=\llbracket {\mathbf {f}}\rrbracket \cdot \delta \hat{{\mathbf {r}}}-\llbracket {\mathbf {f}}\cdot {\mathbf {r}}'\rrbracket \delta \xi , \end{aligned} \end{aligned}$$

(82)

from which the terms in Eqs. (79) and (81) are rewritten as:

$$\begin{aligned} \llbracket \dot{{\mathbf {r}}}\cdot \delta {\mathbf {r}}\rrbracket&=\llbracket \dot{{\mathbf {r}}}\rrbracket \cdot \delta \hat{{\mathbf {r}}}-\llbracket \dot{{\mathbf {r}}}\cdot {\mathbf {r}}'\rrbracket \delta \xi , \end{aligned}$$

(83)

(84)

The term $\llbracket \dot{{\mathbf {r}}}\cdot {\mathbf {r}}'\rrbracket$ can be further transformed by using the relationship:

$$\begin{aligned} \llbracket \dot{{\mathbf {r}}}\cdot \dot{{\mathbf {r}}}\rrbracket +2\,{\dot{\xi }}\llbracket \dot{{\mathbf {r}}}\cdot {\mathbf {r}}'\rrbracket +{\dot{\xi }}^{2}\llbracket {{\mathbf {r}}}'\cdot {\mathbf {r}}'\rrbracket =0, \end{aligned}$$

(85)

which is obtained by manipulating as follows Eq. (4):

$$\begin{aligned} \begin{aligned} \dot{{\mathbf {x}}}\cdot \dot{{\mathbf {x}}}&=\left( \dot{{\mathbf {r}}}_{\pm }+{\dot{\xi }}\,{\mathbf {r}}'_{\pm }\right) \cdot \left( \dot{{\mathbf {r}}}_{\pm }+{\dot{\xi }}\,{\mathbf {r}}'_{\pm }\right) \\&=\dot{{\mathbf {r}}}_{\pm }\cdot \dot{{\mathbf {r}}}_{\pm }+2\,{\dot{\xi }}\,\dot{{\mathbf {r}}}_{\pm }\cdot {\mathbf {r}}'_{\pm }+{\dot{\xi }}^{2}\,{\mathbf {r}}'_{\pm }\cdot {\mathbf {r}}'_{\pm }. \end{aligned} \end{aligned}$$

(86)

By collecting all previous results, the variational principle (71) reads:

(87)

from which Eqs. (14)–(18) are finally derived.

Appendix B: Numerical solution of the integro-differential system

To solve the final Eq. (70), a numerical procedure is adopted, in which the trapezoidal rule for the integral and the forward finite differences for the time-derivatives are adopted. The following positions are first introduced for brevity:

$$\begin{aligned} \begin{aligned}&f\left( t\right) :=-\frac{P\,t^{2}}{2\,\mu },\\&{\mathcal {A}}\left( t,\tau \right) :=K\left( \xi \left( t\right) ,t,\tau \right) +\frac{t-\tau }{\mu },\\&{\mathcal {B}}\left( t,\tau \right) :=2\sum _{k=1}^{N_{e}}\cos \left( \omega _{k}\xi \left( t\right) \right) \sin \left( \omega _{k}\xi \left( \tau \right) \right) \\&\quad \sin \left( \omega _{k}\left( t-\tau \right) \right) . \end{aligned} \end{aligned}$$

(88)

The time interval $\left[ 0,t_{f}\right]$ is divided in $N_{s}>2$ equispaced time sub-intervals of amplitude $\Delta t=t_{f}/N_{s}$. The following notation is used:

$$\begin{aligned} \begin{aligned}&t_{i}=i\,\Delta t,&\tau _{j}=j\,\Delta t,&i,j=0,\,1,\,2,\cdots ,N_{s},\\&R_{yi}=R_{y}\left( t_{i}\right) ,&\xi _{i}=\xi \left( t_{i}\right) ,\\&{\mathcal {A}}_{ij}={\mathcal {A}}\left( t_{i},\tau _{j}\right) ,&{\mathcal {B}}_{ij}={\mathcal {B}}\left( t_{i},\tau _{j}\right) ,\\&f_{i}=f\left( t_{i}\right) ,&D_{i}=D_{i}\left( t_{i}\right) \end{aligned} \end{aligned}$$

(89)

The integral Eq. (70a) is approximated as:

$$\begin{aligned}&\left( \frac{1}{2}{\mathcal {A}}_{10}R_{y0}+\frac{1}{2}{\mathcal {A}}_{11}R_{y1}\right) \Delta t=f_{1},\nonumber \\&\left( \frac{1}{2}{\mathcal {A}}_{i0}R_{y0}+\sum _{j=1}^{i-1}{\mathcal {A}}_{ij}R_{yj}+\frac{1}{2}{\mathcal {A}}_{ii}R_{yi}\right) \Delta t=f_{i},\nonumber \\&i=2,\cdots ,N_{s}, \end{aligned}$$

(90)

and the integro-differential Eq. (70b), by accounting for the initial conditions, as:

$$\begin{aligned} \begin{aligned}&\xi _{0}=0,\\&\xi _{1}={\dot{\xi }}_{0}\,\Delta t,\\&\mu \,\frac{\xi _{2}-2\,\xi _{1}+\xi _{0}}{\Delta t^{2}}=D_{0},\\&\mu \,\frac{\xi _{3}-2\,\xi _{2}+\xi _{1}}{\Delta t^{2}}\\&\quad =D_{1}+R_{y1}\,\left( \frac{1}{2}{\mathcal {B}}_{10}R_{y0}+\frac{1}{2}\mathsf {{\mathcal {B}}}_{11}R_{y1}\right) \Delta t,\\&\mu \,\frac{\xi _{i+2}-2\,\xi _{i+1}+\xi _{i}}{\Delta t^{2}}=D_{i}\\&\qquad +R_{yi}\,\left( \frac{1}{2}\mathsf {{\mathcal {B}}}_{i0}R_{y0}+\sum _{j=1}^{i-1}{\mathcal {B}}_{ij}R_{yj}+\frac{1}{2}\mathsf {{\mathcal {B}}}_{ii}R_{yi}\right) \Delta t,\\&\quad \quad i=2,\cdots ,N_{s}. \end{aligned} \end{aligned}$$

(91)

Equations (90) and (91) can be solved in cascade by following the sequence:

$$\begin{aligned}&\xi _{0}= 0,\nonumber \\&\xi _{1}= {\dot{\xi }}_{0}\,\Delta t,\nonumber \\&R_{y0}= \frac{2\,f_{1}}{{\mathcal {A}}_{10}\,\Delta t},\nonumber \\&\xi _{2}= \frac{D_{0}\Delta t^{2}}{\mu }+2\,\xi _{1}-\xi _{0},\nonumber \\&R_{y1}= \frac{f_{2}}{{\mathcal {A}}_{21}\,\Delta t}-\frac{{\mathcal {A}}_{20}}{2{\mathcal {A}}_{21}}R_{y0},\nonumber \\&\xi _{3}= \frac{D_{1}\Delta t^{2}}{\mu }+2\,\xi _{2}-\xi _{1}+\frac{1}{2\mu }{\mathcal {B}}_{10}R_{y0}R_{y1}\Delta t^{3},\nonumber \\&R_{y\left( i-1\right) }= \frac{f_{i}}{{\mathcal {A}}_{i\left( i-1\right) }\,\Delta t}\nonumber \\&\quad -\frac{1}{{\mathcal {A}}_{i\left( i-1\right) }}\left( \frac{1}{2}{\mathcal {A}}_{i0}R_{y0}+\sum _{j=1}^{i-2}{\mathcal {A}}_{ij}R_{yj}\right) ,\nonumber \\&i=3,\cdots ,N_{s},\nonumber \\&\xi _{i+1}= \frac{D_{\left( i-1\right) }\Delta t^{2}}{\mu }+2\,\xi _{i}-\xi _{i-1}\nonumber \\&\quad +\frac{R_{y\left( i-1\right) }}{\mu }\,\left( \frac{1}{2}\mathsf {{\mathcal {B}}}_{\left( i-1\right) 0}R_{y0}+\sum _{j=1}^{i-2}\mathsf {{\mathcal {B}}}_{\left( i-1\right) j}R_{yj}\right) \Delta t^{3},\nonumber \\&i=3,\cdots ,N_{s}, \end{aligned}$$

(92)

in which ${\mathcal {A}}_{ii}={\mathcal {B}}_{ii}=0$ has been accounted for (i.e., ${\mathcal {A}}\left( t,t\right) ={\mathcal {B}}\left( t,t\right) =0$). In the case of assigned $\xi$-motion, the step relevant to the determination of $\xi _{i}$ is skipped and subsequently utilized for the determination of $D_{i}$.

Appendix C: Massless string

The equation of motion of the massless string is obtained by neglecting the inertia term in Eq. (51):

$$\begin{aligned}&v''+R_{y}\,\delta \left( s-\xi \right) =0. \end{aligned}$$

(93)

Equation (93) admits the solution:

$$\begin{aligned} v\left( s,t\right) ={\left\{ \begin{array}{ll} R_{y}\,\left( 1-\xi \right) \,s &{} 0\le s\le \xi \\ R_{y}\,\left( 1-s\right) \,\xi &{} \xi \le s\le 1 \end{array}\right. }. \end{aligned}$$

(94)

Substitution of Eq. (94) in Eq (54) yields:

$$\begin{aligned} y=v\left( \xi ,t\right) =R_{y}\,\left( 1-\xi \right) \,\xi , \end{aligned}$$

(95)

from which:

$$\begin{aligned} R_{y}=\frac{y}{\left( 1-\xi \right) \,\xi }. \end{aligned}$$

(96)

From Eq. (94), it follows that:

$$\begin{aligned} \left\langle v'\right\rangle =\frac{1}{2}\left( 1-2\,\xi \right) \,R_{y}. \end{aligned}$$

(97)

By using Eqs. (96) and (97), Eqs. (52) and (53) become:

$$\begin{aligned}&\mu \,\ddot{y}+\frac{y}{\left( 1-\xi \right) \,\xi }=-P, \end{aligned}$$

(98)

$$\begin{aligned}&\mu \,\ddot{\xi }+\frac{\left( 2\,\xi -1\right) y^{2}}{2\left( 1-\xi \right) ^{2}\,\xi ^{2}}=D. \end{aligned}$$

(99)

Equations (98) and (99) govern the problem of the massless string. Once such equations are solved for y and $\xi$, the vertical reaction and the displacement field of the string are evaluated via Eqs. (96) and (94), then the nondimensional horizontal displacement, the horizontal reaction and the dynamic tension computed through Eqs. (60), (61) and (62), respectively.

It is worth noting that some coefficients of Eqs. (98) and (99) exhibit singularities at $\xi =0$ or $\xi =1$. This situation does not occur if the string is hanging on two vertical elastic supports, as proven in [24]. Therefore, the springs regularize the mathematical model. To investigate the role of singularities, a numerical analysis (not shown here) was carried out, comparing results of Eqs. (98), (99) and those derived in [24], when a very small compliance of the springs is taken. The two models were found to be in excellent agreement, except for a very narrow layer close to the ends. Therefore, the simpler model was adopted.