TY - JOUR

T1 - A priori estimates for the general dynamic Euler–Bernoulli beam equation

T2 - Supported and cantilever beams

AU - Hasanov, Alemdar

AU - Itou, Hiromichi

N1 - Funding Information:
This research has been supported by the Japan Society for the Promotion of Science (JSPS) through the International Program FY2018 JSPS Invitational Fellowship for Research in Japan (short-term) (No. S18019 ). The authors are grateful to the anonymous referees for careful reading of the paper and for pointing out some minor errors.
Funding Information:
This research has been supported by the Japan Society for the Promotion of Science (JSPS) through the International Program FY2018 JSPS Invitational Fellowship for Research in Japan (short-term) (No. S18019). The authors are grateful to the anonymous referees for careful reading of the paper and for pointing out some minor errors.

PY - 2019/1

Y1 - 2019/1

N2 - This work is a further development of weak solution theory for the general Euler–Bernoulli beam equation ρ(x)utt+μ(x)ut+r(x)uxx xx−(Tr(x)ux)x=F(x,t) defined in the finite dimension domain ΩT≔(0,l)×(0,T)⊂R2, based on the energy method. Here r(x)=EI(x), E>0 is the elasticity modulus and I(x)>0 is the moment of inertia of the cross-section, ρ(x)>0 is the mass density of the beam, μ(x)>0 is the damping coefficient and Tr(x)≥0 is the traction force along the beam. Two benchmark initial boundary value problems with mixed boundary conditions, corresponding to supported and cantilever beams, are analyzed. For the weak and regular weak solutions of these problems a priori estimates are derived under the minimal conditions. These estimates in particular imply the uniqueness of the solutions of both problems.

AB - This work is a further development of weak solution theory for the general Euler–Bernoulli beam equation ρ(x)utt+μ(x)ut+r(x)uxx xx−(Tr(x)ux)x=F(x,t) defined in the finite dimension domain ΩT≔(0,l)×(0,T)⊂R2, based on the energy method. Here r(x)=EI(x), E>0 is the elasticity modulus and I(x)>0 is the moment of inertia of the cross-section, ρ(x)>0 is the mass density of the beam, μ(x)>0 is the damping coefficient and Tr(x)≥0 is the traction force along the beam. Two benchmark initial boundary value problems with mixed boundary conditions, corresponding to supported and cantilever beams, are analyzed. For the weak and regular weak solutions of these problems a priori estimates are derived under the minimal conditions. These estimates in particular imply the uniqueness of the solutions of both problems.

KW - A priori estimates

KW - Euler–Bernoulli beam equation

KW - Uniqueness

KW - Weak and regular weak solution

UR - http://www.scopus.com/inward/record.url?scp=85051650664&partnerID=8YFLogxK

U2 - 10.1016/j.aml.2018.07.038

DO - 10.1016/j.aml.2018.07.038

M3 - Article

AN - SCOPUS:85051650664

VL - 87

SP - 141

EP - 146

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

ER -