9.4 Vibration of plates and shells

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

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• 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 $\nabla^4 w + \frac{\rho h}{D} \frac{\partial^2 w}{\partial t^2} = \frac{q}{D}$ Where $w$ displacement, $\rho$ density, $h$ thickness, $D$ flexural rigidity, $q$ applied load
  • 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 $W_{mn}(x,y) = \sin(\frac{m\pi x}{a})\sin(\frac{n\pi y}{b})$ Where $m,n$ mode numbers, $a,b$ plate dimensions
• 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 $\omega_{mn} = \sqrt{\frac{E}{\rho(1-\nu^2)}} \sqrt{(\frac{m\pi}{L})^2 + (\frac{n}{R})^2}$ Where $E$ Young's modulus, $\nu$ Poisson's ratio, $L$ length, $R$ radius
• 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 $D\nabla^4 w + \rho h \frac{\partial^2 w}{\partial t^2} = q_0 e^{i\omega t}$ Where $q_0$ amplitude of harmonic load, $\omega$ excitation frequency

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