Eur. J. Phy 2015)055035 J Xu et al x(x+)-X"(x2)=mk(xr),(0<x<1) (13) There is a jump at X (') due to the rod carrying a concentrated mass at i'*l If the load is located at the endpoint of x'=l, equations(11) and(12) become X"(1)=0,and-X"(1)=k4X(1) For 0<i'<l, the continuous connection conditions at the both sides of the jump are (x4)=X(x2),X(x4)=X(x),andX"(x)=x”(x2 We use a linear combination of such allowed solutions, expressed as the sum of the cosine wave, hyperbolic cosine wave, sinusoids, and hyperbolic sinusoids of kt, to describe the transverse vib ations of a rod of length I clamped at end of x=0. The solutions of X (1)=A(cosh ki - cos ki)+ B(sinh ki- sin ki),(O<I<A-.(16) X(x)= C cosh kr+ D cos kt+ F sinh ki+ G sin kt,,(x≤x≤1) They satisfy the conditions (11). From the conditions(12), we have k-X.(1)=C cosh k-D cos k F sinh k-G sin k (18) k-X>'(1)=C sinh k +D sin k F cosh k-G cos k (19) From the continuous connection conditions in equation(15)with k'E kr, we have =C cosh k+D cos k'+ F sink+G sin k' (20) A(sinh k'+ sin k)+B(cosh k'-cos k?) =C sinhk'-D sin k+ F cosh k'+G cos k A(cosh k+ cos k)+b(sinh k'+ sin k?) C cosh k'-D cosk'+F sinhk'-g sin k (22) The jump at equation(13)leads to A(sinh k'- sin k)-B(cosh k+ cos k) +C sinh k'+D sin k+ F cosh k-g cos k =k-[A(cosh K- cos k)+ B(sinh k'-sin k)]. Six equations(18)(23)comprise a set of linear homogeneous equations with six coefficient variables(A, B, C, D, F, G), in which the physical variables are m/m and x'/ with the parameter k to be determined. The condition of non-zero solution of equations(18)- (23)is the vanishing determinant. So the eigenvalue equation of k is‴ ′ − ‴ ′ = ′ Xx Xx ( ) + − ( ) ( )( ) ′ < ′< m m ¯¯ ¯ kX x x , 0 ¯ 1 . (13) 4 There is a jump at X x ‴(¯′) due to the rod carrying a concentrated mass at x¯′ ≠ 1. If the load is located at the endpoint of x l ′ = , equations (11) and (12) become ″= − ‴ = ′ X X m m (1) 0, and (1) (1). (14) k X4 For 0 < ′< x¯ 1, the continuous connection conditions at the both sides of the jump are ′ = ′ ′ ′ = ′ ′ ″ ′ = ″ ′ Xx Xx X x X x X x X x () () () ¯ ¯ + −+ − + − () () () , ¯ ¯ , and ¯ ¯ . (15) We use a linear combination of such allowed solutions, expressed as the sum of the cosine wave, hyperbolic cosine wave, sinusoids, and hyperbolic sinusoids of kx¯, to describe the transverse vibrations of a rod of length l clamped at end of x = 0. The solutions of equation (10) are < = − + − ⩽⩽ ′ X x A kx kx B kx kx x x ( ¯) cosh ( ) ( )( ) ¯ cos ¯ sinh ¯ sin ¯ , 0 ¯ ¯− . (16) X x C kx D kx F kx G kx x x >( ¯) cosh = ++ + ¯ cos ¯ sinh ¯ sin ¯, ( ) ¯ ¯ + ′ ⩽ ⩽ 1 . (17) They satisfy the conditions (11). From the conditions (12), we have ″ = −+ − −k X C kD kF kG k > (1) cosh cos sinh sin . (18) 2 ″′ = ++ − −k X C kD kF kG k > (1) sinh sin cosh cos . (19) 3 From the continuous connection conditions in equation (15) with k′≡ ′ kx¯ , we have ′− ′ + ′− ′ = ′+ ′+ ′+ ′ A k kB k k C kD kF kGk (cosh cos ) (sinh sin ) cosh cos sinh sin , (20) ′+ ′ + ′− ′ = ′− ′+ ′+ ′ A k kB k k C kDkF kG k (sinh sin ) (cosh cos ) sinh sin cosh cos , (21) ′+ ′ + ′+ ′ = ′− ′+ ′− ′ A k kB k k C kD kF kGk (cosh cos ) (sinh sin ) cosh cos sinh sin . (22) The jump at equation (13) leads to − ′− ′ − ′+ ′ + ′+ ′+ ′− ′ = ′ ′− ′ + ′− ′ A k kB k k C kDkF kG k k m m A k kB k k (sinh sin ) (cosh cos ) sinh sin cosh cos [ (cosh cos ) (sinh sin )]. (23) Six equations (18)–(23) comprise a set of linear homogeneous equations with six coefficient variables (A,,,,, BCDFG), in which the physical variables are m m′ and x l ′ with the parameter k to be determined. The condition of non-zero solution of equations (18)– (23) is the vanishing determinant. So the eigenvalue equation of k is Eur. J. Phys. 36 (2015) 055035 J Xu et al 6