Explicit solution for vibrating bar with viscous boundaries and internal damper

We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each end, and an internal damper at an arbitrary point along the bar's length. The system is described by four independent parameters and exhibits a variety of behaviors including rigid motion, super stability/instability and zero damping. The solution is obtained by applying the Laplace transform to the equation of motion and computing the Green's function of the transformed problem. This leads to an unconventional eigenvalue-like problem with the spectral variable in the boundary conditions. The eigenmodes of the problem are necessarily complex-valued and are not orthogonal in the usual inner product. Nonetheless, in generic cases we obtain an explicit eigenmode expansion for the response of the bar to initial conditions and external force. For some special values of parameters the system of eigenmodes may become incomplete, or no non-trivial eigenmodes may exist at all. We thoroughly analyze physical and mathematical reasons for this behavior and explicitly identify the corresponding parameter values. In particular, when no eigenmodes exist, we obtain closed form solutions. Theoretical analysis is complemented by numerical simulations, and analytic solutions are compared to computations using finite elements.Comment: 29 pages, 6 figure

Autonomic nervous system dysfunction predicts poor prognosis in patients with mild to moderate tetanus

BACKGROUND: Autonomic nervous system (ANS) dysfunction is present in up to one third of patients with tetanus. The prognostic value of ANS dysfunction is known in severe tetanus but its value is not well established in mild to moderate tetanus. METHODS: Medical records of all patients admitted with tetanus at two academic tertiary care centers in Karachi, Pakistan were reviewed. The demographic, clinical and laboratory data was recorded and analyzed. ANS dysfunction was defined as presence of labile or persistent hypertension or hypotension and sinus tachycardia, tachyarrythmia or bradycardia on EKG. Patients were divided into two groups based on presence of ANS dysfunction (ANS group and non ANS group). Tetanus severity was classified on the basis of Ablett criteria. RESULTS: Ninety six (64 males; 32 females) patients were admitted with the diagnosis over a period of 10 years. ANS group had 31 (32%) patients while non ANS group comprised of 65 (68%) patients. Both groups matched for age, gender, symptom severity, use of tetanus immunoglobulin and antibiotics. Twelve patients in ANS group had mild to moderate tetanus (Ablett I and II) and 19 patients had severe/very severe tetanus (Ablett III and IV). Fifteen (50%) patients in ANS group required ventilation as compared to 28 (45%) in non-ANS group (p = 0.09). Fourteen (47%) patients died in ANS group as compared to 10 (15%) in non ANS group (p= 0.002). Out of those 14 patients died in ANS group, six patients had mild to moderate tetanus and eight patients had severe/ very severe tetanus. Major cause of death was cardiac arrhythmias (13/14; 93%) in ANS group and respiratory arrest (7/10; 70%) in non ANS group. Ten (33%) patients had complete recovery in ANS group while in non ANS group 35(48%) patients had complete recovery (p= 0.05). CONCLUSIONS: ANS dysfunction was present in one third of our tetanus population. 40% patients with ANS dysfunction had only mild to moderate tetanus. ANS dysfunction, irrespective of the need of mechanical ventilation or severity of tetanus, predicted poor outcome

Stability and monotonicity of Lotka–Volterra type operators

In the present paper,we investigate stability of trajectories ofLotka–Volterra (LV) type operators defined in finite dimensional simplex.We prove that any LV type operator is a surjection of the simplex. It is introduced a newclass of LV-type operators, called MLV type ones, and we show that trajectories of the introduced operators converge. Moreover, we show that such kind of operators have totally different behavior than f-monotone LV type operators

Suppression of overshoots and undershoots in nonlinear structural and mechanical systems

In this work, we consider the problem of having a transit in a nonlinear SDOF system from a given rest position to another one, both associated to given and constant values of the external force, without spurious undershoots or overshoots of the solution after the final state is reached. It extends to the nonlinear regime the same problem considered in Udwadia (Acta Mech 231:3157-3182, 2020) in the linear realm. Two different approaches are considered. In the first, which is the more general, the free dynamics of the nonlinear system is not considered, and a very simple solution is obtained, showing how it is robust with respect to perturbations. In the second case, on the other hand, the free dynamics of the system is exploited during the transits, thereby allowing less control parameters to be determined. However, aiming at having a closed form solution, we limit this study to the Duffing equation, although the ideas are general and can be applied to other systems even when closed form solutions are not available