Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

Silverman, Mark P.; Farrah, Joseph

doi:10.4236/wjm.2018.87022

Cited by 4 publications

(5 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even though fishing is a very popular and multi-billion dollar sporting activity, there is very little information available on the physics of the subject. I located only three physics papers [8][9][10], all concerning the complex mathematics involved in projecting a moderately heavy fishing line when fly-fishing, plus several engineering studies [11,12] of the bending of tapered fishing rods.…”

Section: Discussionmentioning

confidence: 99%

Impulse excitation of mechanical resonators: mass–spring systems and fishing rods

Cross

2020

Eur. J. Phys.

View full text Add to dashboard Cite

A mass attached to a spring can be set in motion by applying a force either to the mass or the spring. We consider a problem where a short impulse is applied to the spring in an attempt to project the mass at maximum speed. The problem arises when swinging a flexible golf club or a fishing rod, both of which vibrate at low frequency as a mass–spring system. It is shown that maximum speed of the mass at the end of a spring usually arises when the impulse duration is slightly greater than half of the vibration period of the mass–spring system. If that condition is maintained then the speed will increase if the amplitude of the impulse is increased and/or if the vibration frequency of the mass–spring system is increased. Experimental results are presented for a steel ball attached to a soft spring, and also for a short fishing rod, both being suitable for student experiments.

show abstract

Section: Discussionmentioning

confidence: 99%

Impulse excitation of mechanical resonators: mass–spring systems and fishing rods

Cross

2020

Eur. J. Phys.

View full text Add to dashboard Cite

show abstract

“…This paper reports the theoretical solution of three basic models by which to analyze the reaction forces of a fixed ladder represented as a single Euler-Bernoulli (E-B) beam. The characteristics of an E-B beam pertinent to the present study are as follows [9]: 1) the length of the beam is much greater than the linear dimension of the cross-section;…”

Section: The Fixed Ladder As An Euler-bernoulli Beammentioning

confidence: 99%

“…whose solution is called the elastica [17]. Although an exact solution would be required for a highly flexible beam such as a fishing rod [9], the deflection of the rail of a ladder is sufficiently low that one can neglect the square of the slope in x H x x − for non-negative integer n.…”

Section: Analysis Of a Single-beam Fixed Laddermentioning

confidence: 99%

Reaction Forces on a Fixed Ladder in Static Equilibrium: Analysis and Definitive Experimental Test of the Ladder Problem

Silverman¹

2018

WJM

Self Cite

View full text Add to dashboard Cite

The development of a theoretical model to predict the four equilibrium forces of reaction on a simple ladder of non-adjustable length leaning against a wall has long remained an unresolved matter. The difficulty is that the problem is statically indeterminate and therefore requires complementary information to obtain a unique solution. This paper reports 1) a comprehensive theoretical analysis of the three fundamental models based on treating the ladder as a single Euler-Bernoulli beam, and 2) a detailed experimental investigation of the forces of reaction as a function of applied load and location of load. In contrast to previous untested proposals that the solution to the ladder problem lay in the axial constraint on compression or the transverse constraint on flexure, the experimental outcome of the present work showed unambiguously that 1) the ladder could be modeled the best by a pinned support at the base (on the ground) and a roller support at the top (at the wall), and 2) the only complementary relation needed to resolve the static indeterminacy is the force of friction at the wall. Measurements were also made on the impact loading of a ladder by rapid ascent and descent of a climber. The results obtained were consistent with a simple dynamical model of the ladder as a linear elastic medium subject to a pulse perturbation. The solution to the ladder problem herein presented provides a basis for theoretical extension to other types of ladders. Of particular importance, given that accidents involving ladders in the workplace comprise a significant fraction of all industrial accidents, the theoretical relations reported here can help determine whether a collapsed structure, against which a ladder was applied, met regulatory safety limits or not.

show abstract

“…where the first equality in Equation 4 The coefficient µ is not predicted by the model, but is obtained empirically from the data (i.e. the measured reactions) by a method of visual inspection and adjustment [21] or by standard statistical fitting procedures such as the method of maximum likelihood, method of least squares, or Bayes' theorem [22]. As shown in detail in Part I, the forces of reaction given by Equations (3)-(5) for a constant value of µ satisfactorily accounted for all empirical reactions measured for a wide range of loads P and load locations β for both a fixed ladder and a single E-B beam.…”

Section: Force Of Static Friction On a Ladder Or Beammentioning

confidence: 99%

The Role of Friction in the Static Equilibrium of a Fixed Ladder: Theoretical Analysis and Experimental Test

Silverman¹

2018

WJM

Self Cite

View full text Add to dashboard Cite

In a recent publication the author derived and experimentally tested several theoretical models, distinguished by different boundary conditions at the contacts with horizontal and vertical supports, that predicted the forces of reaction on a fixed (i.e. inextensible) ladder. This problem is statically indeterminate since there are 4 forces of reaction and only 3 equations of static equilibrium. The model that predicted the empirical reactions correctly used a law of static friction to complement the equations of static equilibrium. The present paper examines in greater theoretical and experimental detail the role of friction in accounting for the forces of reaction on a fixed ladder. The reported measurements confirm that forces parallel and normal to the support at the top of the ladder are linearly proportional with a constant coefficient of friction irrespective of the magnitude or location of the load, as assumed in the theoretical model. However, measurements of forces parallel and normal to the support at the base of the ladder are linearly proportional with coefficients that depend sensitively on the location (although not the magnitude) of the load. This paper accounts quantitatively for the different effects of friction at the top and base of the ladder under conditions of usual use whereby friction at the vertical support alone is insufficient to keep the ladder from sliding. A theoretical model is also proposed for the unusual circumstance in which friction at the vertical support can keep the ladder from sliding.

show abstract

Bending of a Tapered Rod: Modern Application and Experimental Test of Elastica Theory

Cited by 4 publications

References 20 publications

Impulse excitation of mechanical resonators: mass–spring systems and fishing rods

Impulse excitation of mechanical resonators: mass–spring systems and fishing rods

Reaction Forces on a Fixed Ladder in Static Equilibrium: Analysis and Definitive Experimental Test of the Ladder Problem

The Role of Friction in the Static Equilibrium of a Fixed Ladder: Theoretical Analysis and Experimental Test

Contact Info

Product

Resources

About