9.4 Vibration of plates and shells

Equations of motion for plates and shells

Fundamental principles and geometry

Plate and shell theories rest on assumptions about how these structures deform. Thin plate/shell theory (Kirchhoff-Love) assumes that normals to the mid-surface remain straight and perpendicular after deformation, and that transverse shear deformation is negligible. Thick plate/shell theory relaxes these assumptions to account for shear effects. Choosing the right theory depends on the thickness-to-span ratio of your structure.

The coordinate systems differ between the two:

• Plates use Cartesian coordinates $(x, y, z)$, where $z$ is the out-of-plane direction
• Shells use curvilinear coordinates that follow the curved geometry (cylindrical $(x, \theta, r)$ or spherical $(r, \theta, \phi)$)

Stress-strain relationships define how the material responds to loading:

• Isotropic materials have uniform properties in all directions (metals like steel or aluminum)
• Anisotropic materials have direction-dependent properties (fiber-reinforced composites, where stiffness along the fiber direction differs from the transverse direction)

Energy methods for equation derivation

The equations of motion for plates and shells are typically derived using energy methods rather than direct force balance, because the geometry makes free-body diagrams unwieldy.

Hamilton's principle states that the actual motion of a system minimizes the time integral of the Lagrangian (kinetic energy minus potential energy) over any time interval. The steps to derive the governing equation are:

1. Write the kinetic energy expression, accounting for mass and velocity distributions across the plate/shell thickness

2. Write the potential (strain) energy expression from the deformation field

3. Form the Lagrangian $L = T - U$ (kinetic minus potential energy)

4. Apply the variational principle $\delta \int_{t_1}^{t_2} L \, dt = 0$

5. Carry out the variation and integrate by parts to obtain the governing PDE and associated boundary conditions

For a thin plate under Kirchhoff-Love theory, this process yields the classical plate equation:

$\nabla^4 w + \frac{\rho h}{D} \frac{\partial^2 w}{\partial t^2} = \frac{q}{D}$

where:

• $w$ = transverse displacement
• $\rho$ = material density
• $h$ = plate thickness
• $D = \frac{Eh^3}{12(1-\nu^2)}$ = flexural rigidity ($E$ is Young's modulus, $\nu$ is Poisson's ratio)
• $q$ = applied transverse load per unit area
• $\nabla^4 = \nabla^2(\nabla^2)$ is the biharmonic operator

For thick plates, higher-order shear deformation theories (such as Mindlin plate theory) add terms to account for transverse shear, which becomes significant when the thickness-to-span ratio exceeds roughly 1/10.

Shell equations of motion

Shell equations are more complex than plate equations because curvature couples in-plane (membrane) and out-of-plane (bending) behavior.

• Membrane effects dominate in thin shells and involve in-plane stretching and compression. These carry loads very efficiently, which is why thin shells can span large distances.
• Bending effects become significant in thicker shells or near boundaries and discontinuities, where the membrane state alone can't satisfy the boundary conditions.

This coupling means that shell equations involve multiple displacement components (axial, circumferential, and radial) linked together. The Donnell-Mushtari-Vlasov (DMV) equations for thin cylindrical shells are one of the simpler shell formulations, but they still consist of coupled PDEs in three displacement variables. More refined theories (Flügge, Sanders) improve accuracy for shorter shells or lower circumferential mode numbers at the cost of added complexity.

Natural frequencies and mode shapes of plates and shells

Free vibration analysis

Free vibration is what happens when a structure oscillates with no external force acting on it. Every plate or shell has an infinite set of natural frequencies, each paired with a mode shape that describes the spatial deformation pattern at that frequency.

The standard solution approach uses separation of variables:

1. Assume the displacement has the form $w(x,y,t) = W(x,y) \cdot T(t)$
2. Substitute into the homogeneous governing equation (set $q = 0$)
3. Separate the spatial and temporal parts. The time function satisfies $\ddot{T} + \omega^2 T = 0$, giving harmonic oscillation at frequency $\omega$
4. The spatial function $W(x,y)$ satisfies an eigenvalue problem whose solutions depend on the boundary conditions
5. Apply boundary conditions to obtain the characteristic equation, whose roots are the natural frequencies $\omega_{mn}$
6. The corresponding spatial solutions $W_{mn}(x,y)$ are the mode shapes

Plate vibration characteristics

Boundary conditions have a major influence on both natural frequencies and mode shapes:

• Simply supported edges allow rotation but prevent translation ($w = 0$, moment $= 0$)
• Clamped edges restrict both rotation and translation ($w = 0$, slope $= 0$)
• Free edges have no constraints (moment $= 0$, shear $= 0$)

Clamped boundaries raise natural frequencies compared to simply supported ones, because they add stiffness. Free edges lower them.

For a simply supported rectangular plate with dimensions $a \times b$, the mode shapes and natural frequencies have clean closed-form expressions:

$W_{mn}(x,y) = \sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right)$

$\omega_{mn} = \pi^2 \left[\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2\right] \sqrt{\frac{D}{\rho h}}$

where $m$ and $n$ are positive integers representing the number of half-waves in the $x$ and $y$ directions. The $(1,1)$ mode is the fundamental (lowest frequency).

Key trends to remember:

• Increasing aspect ratio ($a/b$) generally lowers the fundamental frequency because the plate becomes more flexible in the longer direction
• Thicker plates have higher natural frequencies, since $D$ scales with $h^3$ while mass scales with $h$, so $\omega \propto h$
• Stiffer materials (higher $E$) increase natural frequencies through the flexural rigidity $D$

Shell vibration analysis

Shell vibration is richer than plate vibration because modes involve axial ($m$), circumferential ($n$), and radial components simultaneously. For a simply supported cylindrical shell, a simplified natural frequency expression is:

$\omega_{mn} = \sqrt{\frac{E}{\rho(1-\nu^2)}} \sqrt{\left(\frac{m\pi}{L}\right)^2 + \left(\frac{n}{R}\right)^2}$

where $L$ is the shell length and $R$ is the radius. Note that this is an approximate expression; more accurate formulations account for the coupling between membrane and bending stiffness, which causes the lowest natural frequency to occur at an intermediate circumferential mode number $n$, not necessarily $n = 1$.

• Increased curvature (smaller $R$) generally raises natural frequencies because curvature adds membrane stiffness
• Boundary conditions affect the frequency spectrum significantly. A clamped-free cylindrical shell behaves very differently from a simply supported one, especially for low circumferential modes

Forced vibration response of plates and shells

Forced vibration fundamentals

When an external time-varying load acts on a plate or shell, the structure undergoes forced vibration. Understanding this response is critical for predicting vibration amplitudes, stress levels, and fatigue life under service conditions.

For a plate subjected to harmonic excitation, the equation of motion becomes:

$D\nabla^4 w + \rho h \frac{\partial^2 w}{\partial t^2} = q_0 e^{i\omega t}$

where $q_0$ is the load amplitude and $\omega$ is the excitation frequency. The goal is to find the steady-state displacement $w(x,y,t)$.

Modal analysis and frequency response

Modal analysis is the standard approach for solving forced vibration problems. It works by expanding the response as a sum of the free-vibration mode shapes, converting one PDE into a set of decoupled ODEs.

The procedure:

1. Solve the free vibration problem to obtain natural frequencies $\omega_{mn}$ and mode shapes $W_{mn}$
2. Expand the displacement as $w(x,y,t) = \sum_{m,n} W_{mn}(x,y) \cdot \eta_{mn}(t)$, where $\eta_{mn}(t)$ are modal coordinates
3. Substitute into the forced equation and use the orthogonality of mode shapes to decouple the equations
4. Solve each decoupled modal equation independently (each is a single-DOF forced oscillator)
5. Sum the modal contributions to get the total response

Frequency response functions (FRFs) describe how the system amplifies or attenuates input at each frequency:

• Resonance occurs when the excitation frequency $\omega$ matches a natural frequency $\omega_{mn}$, producing large response amplitudes
• Anti-resonance is a frequency where the response reaches a local minimum, often occurring between adjacent resonances

Damping plays a critical role near resonance:

• It limits the peak response amplitude (without damping, resonance amplitude would be infinite)
• It shifts resonant frequencies slightly downward
• Common damping mechanisms include viscous damping (velocity-proportional), structural/hysteretic damping (displacement-proportional but in phase with velocity), and acoustic radiation damping (energy lost to sound)

Dynamic load response analysis

Real structures encounter various types of dynamic loads, each requiring different analysis approaches:

• Point loads (concentrated forces) excite many modes and produce localized stress concentrations
• Distributed loads (uniform or varying pressure) tend to excite lower-order modes more strongly
• Impact loads (short-duration impulses) excite a broad frequency range and require transient analysis

Plates and shells also radiate sound when they vibrate. Sound radiation efficiency depends on the mode shape and frequency: modes with wavelengths longer than the acoustic wavelength in the surrounding fluid radiate sound efficiently, while shorter-wavelength modes are poor radiators. This distinction matters in noise control applications.

Approximate methods for plate and shell vibration problems

Rayleigh-Ritz and Galerkin methods

Closed-form solutions exist only for simple geometries and boundary conditions. For everything else, approximate methods are needed.

The Rayleigh-Ritz method estimates natural frequencies and mode shapes:

1. Choose a set of admissible functions that satisfy the geometric (essential) boundary conditions
2. Assume the displacement as a linear combination: $w \approx \sum c_i \phi_i(x,y)$
3. Substitute into the energy expressions (kinetic and potential energy)
4. Minimize the Rayleigh quotient with respect to the unknown coefficients $c_i$
5. This yields a matrix eigenvalue problem $[K]\{c\} = \omega^2 [M]\{c\}$, where $[K]$ and $[M]$ are stiffness and mass matrices

The accuracy depends heavily on how well your trial functions approximate the true mode shapes. Polynomial or trigonometric functions are common choices.

The Galerkin method follows a similar philosophy but works directly with the governing differential equation rather than energy expressions:

1. Assume a trial solution as a series of basis functions
2. Substitute into the governing PDE to form a residual (the equation won't be satisfied exactly)
3. Require the residual to be orthogonal to each basis function (weighted residual approach)
4. This produces a system of algebraic equations for the unknown coefficients

Galerkin's method applies to both linear and nonlinear problems, making it more versatile than Rayleigh-Ritz for certain applications.

Numerical discretization techniques

Finite difference methods replace continuous derivatives with discrete approximations on a grid. For example, a central difference scheme approximates $\frac{\partial^2 w}{\partial x^2}$ at grid point $i$ as $\frac{w_{i+1} - 2w_i + w_{i-1}}{\Delta x^2}$. This converts the PDE into a system of algebraic equations. The method is straightforward to implement but struggles with irregular geometries and complex boundary conditions.

The finite element method (FEM) is the most widely used approach for practical plate and shell vibration problems:

1. Divide the structure into small elements (triangular or quadrilateral for plates/shells)
2. Define displacement interpolation (shape functions) within each element
3. Compute element stiffness $[k_e]$ and mass $[m_e]$ matrices from the energy expressions
4. Assemble element matrices into global stiffness $[K]$ and mass $[M]$ matrices
5. Apply boundary conditions
6. Solve the eigenvalue problem $[K]\{\phi\} = \omega^2 [M]\{\phi\}$ for natural frequencies and mode shapes

FEM handles arbitrary geometries, mixed boundary conditions, and material variations with ease, which is why it dominates in industry.

Advanced analysis techniques

For large-scale models (thousands or millions of degrees of freedom), solving the full eigenvalue problem becomes computationally expensive. Model reduction techniques address this:

• Component mode synthesis (CMS) divides a large structure into substructures, solves each one separately, then couples them through interface conditions. This dramatically reduces the problem size while preserving accuracy for the frequency range of interest.
• The reduced model combines fixed-interface normal modes (vibration modes of each substructure with interfaces fixed) and constraint modes (static deformations due to unit displacements at interface DOFs).

When evaluating any approximate method, consider:

• Convergence: Does the solution improve as you refine the mesh or add more trial functions? A convergence study (progressively refining the model) is essential for verifying accuracy.
• Computational cost: FEM with fine meshes gives high accuracy but requires significant computation time and memory. Rayleigh-Ritz with a few well-chosen functions can give good estimates of the first several natural frequencies at a fraction of the cost.
• The right method depends on the problem: quick design estimates favor Rayleigh-Ritz, while detailed stress analysis of complex geometries requires FEM.