Highly efficient continuous bistable nonlinear energy sink composed of a cantilever beam with partial constrained layer damping

Abstract

This paper focuses on the transient nonlinear dynamics and targeted energy transfer (TET) of a Bernoulli–Euler beam coupled to a continuous bistable nonlinear energy sink (NES). This NES comprises a cantilever beam with the partial constrained layer damping (PCLD) and an end mass controlled by a nonlinear magnetostatic interaction force. The theoretical model of the nonlinear system is built based on the Lagrange equations and assumed-modes expansion method. A new parameter system damping ratio is proposed to evaluate the TET efficiencies. Impact experiments are carried out to verify the theoretical model and mechanisms. The results show that the bistable NES can achieve high and strongly robust TET efficiencies under broad-range impacts. The shear modulus of the viscoelastic layer, the length of the PCLD and the end mass have significant influences on TET efficiencies. Analyses of the TET mechanisms in the bistable NES show the following: steady transition of the stable state is an important reason for maintaining high TET efficiencies; nonlinear beatings can occur in high-frequency, fundamental and long-period subharmonic branches; and resonance captures featuring fundamental and subharmonic also help achieve rapid energy dissipation.

Appendix

The kinetic energies of different parts can be expressed as

$$\begin{aligned}&T_p =\frac{1}{2}\int _0^L {\left[ \rho _p A_p +\sum _i {m_{pi} \delta \left( x-L_{xi} \right) } \right] \left( {\frac{\partial y\left( x,t\right) }{\partial t}} \right) ^{2}{\text{ d }}x}\\&T_b =\frac{1}{2}\int _0^{L_b} {\rho _b A_b \left[ {\left( {\frac{\partial w_r }{\partial t}+{\dot{\varphi }}+x\frac{\partial ^{2}y\left( L_0, t\right) }{\partial x\partial t}} \right) ^{2}+\left( {\frac{\partial u_b }{\partial t}} \right) ^{2}} \right] {\text{ d }}x}\\&T_v +T_c =\frac{1}{2}\left( \rho _v A_v +\rho _c A_c \right) \\&\qquad \quad \qquad \cdot \int _{x_1 }^{x_2} {\left( {\frac{\partial w_r }{\partial t}+{\dot{\varphi }}+x\frac{\partial ^{2}y\left( L_0, t\right) }{\partial x\partial t}} \right) ^{2}{\text{ d }}x} \\&\qquad \quad \qquad +\, \frac{1}{2}\int _{x_1 }^{x_2} {\left( {\rho _v A_v \left( {\frac{\partial u_v }{\partial t}} \right) ^{2}+\rho _c A_c \left( {\frac{\partial u_c }{\partial t}} \right) ^{2}} \right) {\text{ d }}x} \\&T_m =\frac{1}{2}m_t \left( {\frac{\partial w_r \left( L_b, t\right) }{\partial t}+{\dot{\varphi }}+L_b \frac{\partial ^{2}y\left( L_0, t\right) }{\partial x\partial t}} \right) ^{2} \\&\qquad \quad +\,\frac{1}{2}m_t \left( {\frac{\partial u_b \left( L_b, t\right) }{\partial t}} \right) ^{2}+\frac{1}{2}J_t \left( {\frac{\partial ^{2}w_r \left( L_b, t\right) }{\partial x\partial t}} \right) ^{2} \\ \end{aligned}$$

where $J_t$ is the end mass’s moment of inertia relative to the axis oz in Fig. 2b, $J_t ={m_t R_1^2 }/4+{m_t d_1^2 }/{12}$.

Table 4 Parameters of restoring forces in Group 2

Table 5 Parameters for Group 3

The elements in the generalized mass matrix and generalized complex stiffness matrix are listed below.

$$\begin{aligned}&\mathbf{M}_{\mathrm{e}} =\left[ {{ \begin{array}{ccccc} {\mathbf{M}_{\phi \phi } +\mathbf{M}_{c\phi \phi }}&{} {\mathbf{M}_{\phi \eta }}&{} 0&{} 0 \\ {\mathbf{M}_{\phi \eta }^\mathrm{T}}&{} {\mathbf{M}_{\eta \eta }}&{} {\mathbf{M}_{\eta \xi }}&{} {\mathbf{M}_{\eta \alpha }} \\ 0&{} {\mathbf{M}_{\eta \xi }^\mathrm{T}}&{} {\mathbf{M}_{\xi \xi }}&{} {\mathbf{M}_{\xi \alpha }} \\ 0&{} {\mathbf{M}_{\eta \alpha }^\mathrm{T}}&{} {\mathbf{M}_{\xi \alpha }^\mathrm{T}}&{} {\mathbf{M}_{\alpha \alpha }} \\ \end{array}}} \right] \\&\mathbf{K}_{\mathrm{e}} =\left[ {{ \begin{array}{ccccc} {\mathbf{K}_{\phi \phi }}&{} {\mathbf{K}_{\phi \eta }}&{} 0&{} 0 \\ {\mathbf{K}_{\phi \eta }^\mathrm{T}}&{} {\mathbf{K}_{\eta \eta }}&{} {\mathbf{K}_{\eta \xi }}&{} {\mathbf{K}_{\eta \alpha }} \\ 0&{} {\mathbf{K}_{\eta \xi }^\mathrm{T}}&{} {\mathbf{K}_{\xi \xi }}&{} {\mathbf{K}_{\xi \alpha }} \\ 0&{} {\mathbf{K}_{\eta \alpha }^\mathrm{T}}&{} {\mathbf{K}_{\xi \alpha }^\mathrm{T}}&{} {\mathbf{K}_{\alpha \alpha }} \\ \end{array}}} \right] \end{aligned}$$

$$\begin{aligned}&M_{\phi \phi , ij} =\int _0^L {\rho _p A_p Y_i Y_j} {\text{ d }}x+\sum _k {m_{pk} Y_i (L_k )Y_j (L_k )}, \\&\quad i,j=1,2\ldots n_p \\&M_{c\phi \phi , ij} =\int _0^{L_b} {\rho _b A_b [Y_i (L_0 )+xY_i^{\prime }(L_0 )]} \\&\quad \cdot \, [Y_j (L_0 )+xY_j^{\prime }(L_0 )]dx+(\rho _c A_c +\rho _v A_v ) \\&\quad \cdot \int _{x_1 }^{x_2} {[Y_i (L_0 )+xY_i^{\prime }(L_0 )][Y_j (L_0 )+xY_j^{\prime }(L_0 )]} {\text{ d }}x \\&\quad +\, m_t [Y_i (L_0 )+L_b Y_i^{\prime }(L_0 )][Y_j (L_0 )+L_b Y_j^{\prime }(L_0 )], \\&\quad i,j=1,2\ldots n_p \\ \end{aligned}$$

$$\begin{aligned}&\begin{array}{l} M_{\phi \eta , ij} =\displaystyle \int _0^{L_b} {\rho _b A_b [Y_i (L_0 )+xY_i^{\prime }(L_0 )]W_{rj} (x)} {\text{ d }}x \\ \quad +\displaystyle \int _{x_1 }^{x_2} {(\rho _v A_v +\rho _c A_c )[Y_i (L_0 )+\,xY_i^{\prime }(L_0 )]W_{rj} (x){\text{ d }}x} \\ \quad +\, m_t [Y_i (L_0 )+L_b Y_i^{\prime }(L_0 )]W_{rj} (L_b ), \\ \quad i=1,2\ldots n_p, j=1,2\ldots n_w \\ \end{array}\\&M_{\eta \eta , ij} =\int _0^{L_b} {\rho _b A_b W_{ri} W_{rj} {\text{ d }}x}\\&\quad +\int _{x_1 }^{x_2} {(\rho _v A_v +\rho _c A_c )W_{ri} W_{rj} {\text{ d }}x} \\&\quad \cdot \int _{x_1 }^{x_2} {\rho _v A_v \frac{(h_c -h_b )^{2}}{16}{W}_{ri}^{\prime } {W}_{rj}^{\prime } {\text{ d }}x} \\&\quad +\, m_t W_{ri} (L_b )W_{rj} (L_b )+J_t {W}_{ri}^{\prime } (L_b ){W}_{rj}^{\prime } (L_b ),\; \\&\quad i,j=1,2\ldots n_w \\&M_{\eta \xi , ij} =\frac{1}{8}\int _{x_1 }^{x_2} {\rho _v A_v (h_c -h_b ){W}_{ri}^{\prime } U_{bj} {\text{ d }}x}, \\&\quad i=1,2\ldots n_w, j=1,2\ldots n_b \\&M_{\eta \alpha , ij} =\frac{1}{8}\int _{x_1 }^{x_2} {\rho _v A_v (h_c -h_b ){W}_{ri}^{\prime } U_{cj} {\text{ d }}x,\quad } \\&\quad i=1,2\ldots n_w, j=1,2\ldots n_c \\&M_{\xi \xi , ij} =\int _0^{L_b} {\rho _b A_b U_{bi} U_{bj} {\text{ d }}x} \\&\quad +\, \frac{1}{4}\int _{x_1 }^{x_2} {\rho _v A_v U_{bi} U_{bj} {\text{ d }}x} +m_t U_{bi} (L_b )U_{bj} (L_b ), \\&\quad i,j=1,2\ldots n_b \\&M_{\xi \alpha , ij} =\frac{1}{4}\int _{x_1 }^{x_2} {\rho _v A_v U_{bi} U_{cj} {\text{ d }}x}, \\&\quad i=1,2\ldots n_b, j=1,2\ldots n_c \\ \end{aligned}$$

$$\begin{aligned}&M_{\alpha \alpha , ij} =\int _{x_1 }^{x_2} {\left( \rho _c A_c +\frac{1}{4}\rho _v A_v \right) U_{ci} U_{cj} {\text{ d }}x}, \\&\quad i,j=1,2\ldots n_c K_{\phi \phi , ij} =\int _0^L {E_p I_p Y_i^{\prime \prime }Y_j^{\prime \prime }} {\text{ d }}x\\&\quad -\,\beta _1 \left( Y_{\Delta i} +L_b {Y}_{i}^{\prime } \right) \left( Y_{\Delta j} +L_b {Y}_{j}^{\prime } \right) ,\; \\&\quad i,j=1,2\ldots n_p \\&K_{\phi \eta , ij} =-\beta _1 \left( Y_{\Delta i} +L_b {Y}_{i}^{\prime } \right) W_{rj} (L_b ), \\&\quad i=1,2\ldots n_p, j=1,2\ldots n_w \\&K_{\eta \eta , ij} =\int _0^{L_b} {E_b I_b {W}_{ri}^{\prime \prime } {W}_{rj}^{\prime \prime }} {\text{ d }}x+\int _{x_1 }^{x_2} {E_c I_c {W}_{ri}^{\prime \prime } {W}_{rj}^{\prime \prime }} {\text{ d }}x \\&\quad +\int _{x_1 }^{x_2} {\frac{G_v A_v }{h_v^2 }h_t^2 {W}_{ri}^{\prime } {W}_{rj}^{\prime }} {\text{ d }}x-\beta _1 W_{ri} (L_b )W_{rj} (L_b ),\; \\&\quad i,j=1,2\ldots n_w \\&K_{\eta \xi , ij} =-\int _{x_1 }^{x_2} {\frac{G_v A_v }{h_v^2 }h_t {W}_{ri}^{\prime } U_{bj}} {\text{ d }}x, \\&\quad i=1,2\ldots n_w, j=1,2\ldots n_b \\&K_{\eta \alpha , ij} =\int _{x_1 }^{x_2} {\frac{G_v A_v }{h_v^2 }h_t {W}_{ri}^{\prime } U_{cj}} {\text{ d }}x,\\&\quad i=1,2\ldots n_w, j=1,2\ldots n_c \\&K_{\xi \xi , ij} =\int _0^{L_b} {E_b A_b {U}_{bi}^{\prime } {U}_{bj}^{\prime }} {\text{ d }}x+\int _{x_1 }^{x_2} {\frac{G_v A_v }{h_v^2 }U_{bi} U_{bj}} {\text{ d }}x,\; \\&\quad i,j=1,2\ldots n_b \\&K_{\xi \alpha , ij} =-\int _{x_1 }^{x_2} {\frac{G_v A_v }{h_v^2 }U_{bi} U_{cj}} {\text{ d }}x, \\&\quad i=1,2\ldots n_b, j=1,2\ldots n_c \\&K_{\alpha \alpha , ij} =\int _{x_1 }^{x_2} {E_c A_c {U}_{ci}^{\prime } {U}_{cj}^{\prime }} {\text{ d }}x+\int _{x_1 }^{x_2} {\frac{G_v A_v }{h_v^2 }U_{ci} U_{cj}} {\text{ d }}x, \\&\quad i,j=1,2\ldots n_c \end{aligned}$$

$$\begin{aligned}&P_i =\left\{ {\begin{array}{l} (Y_{\Delta i} +L_b {Y}_i^{\prime } )\quad \quad \quad for\;\;\;i=1,2\ldots n_p \\ W_{ri} (L_b )\quad \;for\;\;\;i=n_p +1,\ldots n_p +n_w \\ 0\quad \quad others \\ \end{array}} \right. \\&\mathbf{M}_{\mathrm{n}} =\left[ {{ \begin{array}{ccc} {\mathbf{M}_{\eta \eta }}&{} {\mathbf{M}_{\eta \xi }}&{} {\mathbf{M}_{\eta \alpha }} \\ {\mathbf{M}_{\eta \xi }^\mathrm{T}}&{} {\mathbf{M}_{\xi \xi }}&{} {\mathbf{M}_{\xi \alpha }} \\ {\mathbf{M}_{\eta \alpha }^\mathrm{T}}&{} {\mathbf{M}_{\xi \alpha }^\mathrm{T}}&{} {\mathbf{M}_{\alpha \alpha }} \\ \end{array}}} \right] , \\&\mathbf{K}_{\mathrm{n}} =\left[ {{ \begin{array}{ccc} {\mathbf{K}_{\eta \eta }}&{} {\mathbf{K}_{\eta \xi }}&{} {\mathbf{K}_{\eta \alpha }} \\ {\mathbf{K}_{\eta \xi }^\mathrm{T}}&{} {\mathbf{K}_{\xi \xi }}&{} {\mathbf{K}_{\xi \alpha }} \\ {\mathbf{K}_{\eta \alpha }^\mathrm{T}}&{} {\mathbf{K}_{\xi \alpha }^\mathrm{T}}&{} {\mathbf{K}_{\alpha \alpha }} \\ \end{array}}} \right] \end{aligned}$$