Plates and shells are crucial structural elements in engineering, from aircraft fuselages to pressure vessels. Their vibration behavior is complex, involving intricate interactions between geometry, material properties, and boundary conditions.
Understanding plate and shell vibrations is essential for designing safe and efficient structures. This topic explores the equations of motion, natural frequencies, mode shapes, and forced vibration responses of these continuous systems, providing tools for analysis and design.
Equations of motion for plates and shells
Fundamental principles and geometry
Plate and shell theory assumptions and limitations for thin and thick structures guide vibration analysis
Geometry and coordinate systems define structure behavior
Plates use Cartesian coordinates (x, y, z)
Shells employ curvilinear coordinates (cylindrical or spherical)
Stress-strain relationships and constitutive equations characterize material behavior
Isotropic materials have uniform properties in all directions
Anisotropic materials exhibit direction-dependent properties (fiber-reinforced composites)
Energy methods for equation derivation
Hamilton's principle and Lagrange's equations derive equations of motion
Hamilton's principle minimizes the time integral of the Lagrangian (kinetic energy minus potential energy)
Lagrange's equations use generalized coordinates to describe system motion
Kinetic and potential energy expressions formulated for vibrating plates and shells
Kinetic energy accounts for mass and velocity distributions
Potential energy includes strain energy due to deformation
Governing partial differential equations derived for plate vibration
Classical plate equation (Kirchhoff-Love theory) for thin plates
∇ 4 w + ρ h D ∂ 2 w ∂ t 2 = q D \nabla^4 w + \frac{\rho h}{D} \frac{\partial^2 w}{\partial t^2} = \frac{q}{D} ∇ 4 w + D ρ h ∂ t 2 ∂ 2 w = D q
Higher-order shear deformation theories account for transverse shear effects in thick plates
Shell equations of motion
Cylindrical and spherical shell equations consider membrane and bending effects
Membrane effects dominate in thin shells (in-plane stretching)
Bending effects become significant in thicker shells
Shell equations typically more complex than plate equations due to curvature
Coupling between in-plane and out-of-plane deformations
Example: Donnell-Mushtari-Vlasov (DMV) equations for thin cylindrical shells
Natural frequencies and mode shapes of plates and shells
Free vibration analysis
Free vibration describes natural oscillations without external forces
Natural frequencies represent resonant frequencies of the structure
Mode shapes describe deformation patterns at each natural frequency
Separation of variables technique solves equations of motion
Assumes solution in the form of $w(x,y,t) = W(x,y)T(t)$
Spatial function $W(x,y)$ determines mode shape
Time function $T(t)$ describes harmonic oscillation
Characteristic equation formulated to determine natural frequencies
Derived from boundary conditions and governing equations
Solutions yield eigenvalues (natural frequencies) and eigenvectors (mode shapes)
Plate vibration characteristics
Boundary conditions influence natural frequencies and mode shapes
Simply supported edges allow rotation but no translation
Clamped edges restrict both rotation and translation
Free edges have no constraints
Mode shape functions derived for various boundary conditions
Example: Simply supported rectangular plate mode shape
W m n ( x , y ) = sin ( m π x a ) sin ( n π y b ) W_{mn}(x,y) = \sin(\frac{m\pi x}{a})\sin(\frac{n\pi y}{b}) W mn ( x , y ) = sin ( a mπ x ) sin ( b nπ y )
Aspect ratio, thickness, and material properties affect vibration behavior
Increasing aspect ratio (length/width) generally lowers natural frequencies
Thicker plates have higher natural frequencies due to increased stiffness
Stiffer materials (higher Young's modulus) increase natural frequencies
Shell vibration analysis
Cylindrical shell natural frequencies and mode shapes determined analytically
Axial, circumferential, and radial modes considered
Example: Natural frequency for simply supported cylindrical shell
ω m n = E ρ ( 1 − ν 2 ) ( m π L ) 2 + ( n R ) 2 \omega_{mn} = \sqrt{\frac{E}{\rho(1-\nu^2)}} \sqrt{(\frac{m\pi}{L})^2 + (\frac{n}{R})^2} ω mn = ρ ( 1 − ν 2 ) E ( L mπ ) 2 + ( R n ) 2
Curvature and boundary conditions influence shell vibration characteristics
Increased curvature generally raises natural frequencies
Boundary conditions affect mode shapes and frequency spectrum
Forced vibration response of plates and shells
Forced vibration fundamentals
Forced vibration occurs when external excitation acts on the structure
Crucial for understanding structural response to dynamic loads
Determines vibration amplitudes and stress levels in service
Equations of motion for plates and shells under harmonic excitation
Example: Forced vibration of a plate
D ∇ 4 w + ρ h ∂ 2 w ∂ t 2 = q 0 e i ω t D\nabla^4 w + \rho h \frac{\partial^2 w}{\partial t^2} = q_0 e^{i\omega t} D ∇ 4 w + ρ h ∂ t 2 ∂ 2 w = q 0 e iω t
Modal analysis and frequency response
Modal analysis techniques determine forced response
Decomposes response into contributions from each mode
Utilizes orthogonality properties of mode shapes
Frequency response functions (FRFs) characterize system behavior
Describe amplitude and phase of response relative to input
Resonance occurs when excitation frequency matches natural frequency
Anti-resonance represents minimum response amplitude
Damping effects on forced vibration response
Reduces vibration amplitudes, especially near resonance
Shifts resonant frequencies slightly
Types include viscous, structural, and acoustic radiation damping
Dynamic load response analysis
Steady-state and transient responses to various dynamic loads
Point loads (concentrated forces)
Distributed loads (pressure distributions)
Impact loads (short-duration impulses)
Vibration transmission and sound radiation characteristics
Structure-borne vibration propagation through plates and shells
Sound radiation efficiency depends on mode shapes and frequencies
Approximate methods for plate and shell vibration problems
Rayleigh-Ritz and Galerkin methods
Rayleigh-Ritz method approximates natural frequencies and mode shapes
Assumes mode shapes as linear combinations of admissible functions
Minimizes energy functional to determine coefficients
Example: Rectangular plate with polynomial trial functions
Galerkin method solves forced vibration problems
Expands solution in terms of basis functions
Minimizes residual error in governing equation
Applicable to both linear and nonlinear problems
Numerical discretization techniques
Finite difference methods discretize plate and shell equations
Replaces derivatives with difference approximations
Results in system of algebraic equations
Example: Central difference scheme for plate bending
Finite element method (FEM) models complex structures
Divides structure into small elements (triangular or quadrilateral)
Interpolates displacements within elements
Assembles global stiffness and mass matrices
Solves eigenvalue problem for natural frequencies and mode shapes
Advanced analysis techniques
Model reduction techniques improve computational efficiency
Component mode synthesis (CMS) for large systems
Reduces degrees of freedom while maintaining accuracy
Combines static and dynamic modes of substructures
Accuracy and efficiency evaluation of approximate methods
Convergence studies assess solution accuracy
Computational time and memory requirements compared
Trade-offs between accuracy and efficiency considered for practical applications