Aalto University School of Engineering

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1 Aalto University School of Engineering Kul Ship Structural Design (P) Lecture 8 - Local and Global Vibratory Response

2 Kul Ship Structures Response Lecture 5: Tertiary Response: Bending of plates and stiffeners, σ 3 Design Philosophy Loads Lectures Lecture 6: Bending of web frames, girders and grillages, σ 2 Response Lectures Strength Lectures Lecture 7: Hull girder bending, torsion and shear, σ 1 Lecture 8: Ship vibrations, σ 1 σ 3

3 Contents The aim is to understand how the vibrations are controlled in ship design and what are the main sources of them Vibration response Dynamic response of a ship Vibration characteristics Response for impact load, i.e. unit inpulse Excitation of vibration Machinery Springing and whipping Vibration measurements Literature 1. Clough, R.W. & Penzien, J.; Dynamics of Structures. McGraw-Hill Second Ed. 2. Jensen, J.J., Load and global response of ships, DTU, May Hakala, M, Application of the finite element method to fluid-structure interaction in ship vibration. TKK Konetekniikan osasto väitöskirja, VTT, Research report ), 4. Hughes, O.F.; Ship Structural Design. SNAME, Kosonmaa, L. Jäänmurtokeulaan kohdistuvat iskumaiset aaltokuormat. Diplomityö, TKK/ Konetekniikan osasto, 1996.I 6. Matusiak, J. Laivahydrodynamiikan perusteet. 7. Olkiluoto, P., et. al, Structural Design of an Aluminium Missile Boat. FAST 91, Trondheim.

4 Weekly Exercise Exercise 8: Vibratory Response - Given :00, Return :00 Define the measures to control vibratory levels in your ship. Calculate the eigenfrequencies for plates, stiffeners and girders using basic plate and beam theory Report and discuss the work. Longitudinals and plate Eigenfrequency t [mm] Profile Added Mass [kg/m 2 ] Profile [Hz] Stress in Plate [Hz] Deck 1 5 HP100x

5 Vibration Response Vibrations may occur due to various excitations Machinery and systems Wave-induced Global hull girder vibrations Vertical bending Horizontal bending Torsion Longitudinal Local vibrations Decks and bulkheads Superstructure Etc. Important issues on ship vibrations Mass and stiffness distribution Added mass of vibrating water Effective mass of cargo (cars with suspension, dead load, liquid, etc) Damping FE or analytical methods are used for the analysis The aim is to omit the vibrations by making the design not to coincide the eigenfrequency with the wake frequency

6 Hull Girder Eigenfrequencies and -modes Flexible hull girder (rigid body + deformation modes) [M] { } + [C] {y} + [K] {y} = {F(t)} Displacement {y} can be written as product of eigenmode [φ(x)] and generalized coordinate {p(t)} as {y(x,t)}= [φ(x)] {p(t)} Undamped beam, to obtain eigenfrequencies ω i and shapes φ i we formulate eigenvalue problem: ([ K] ω 2 [ M ]) φ [ K] ω 2 [ M ] = 0. { } = 0.

7 Superposition of Eigenmodes Substitution of {y} as product of eigenmodes [φ(x)] and generalized coordinate {p(t)} and its derivatives gives: [ M][ φ ( x )]{!! p( t) } + [ C][ φ( x) ]{ p! ( t) } + [ K][ φ( x) ]{ p( t) } = { F( t) } Multiplying from left with eigenmode i transpose gives T T T T { φ } [ M][ φ( x) ]{!! p( t) } + { φ } [ C][ φ( x) ]{ p! ( t) } + { φ } [ K][ φ( x) ]{ p( t) } = { φ } { F( t) } i Due to orthogonality T { φ } [M]{ φ } = 0 i T { φ } [C]{ φ } = 0, when i j i T { φ } [K]{ φ } = 0 i We can simplify this to j j j From which the generalized coordinate can be solved The modes are now uncoupled and can be solved. i {φ i } T [M] {φ i } p i + {φ i } T [C] {φ i } p i + {φ i } T [K] {φ i } p i = {φ i } T {F(t)} i i

8 Hull Girder Dynamic Response We can simplify the hull girder response by considering the orthogonality of the eigenmodes p i + 2 ξ i ω i p i + ω 2 i p i = f i (t) M there is no coupling and the i number of equations is the number of eigenmodes to be solved Generalized mass, damping, stiffness and load are M i = {φ i } T [M] {φ i } C i = {φ i } T [C] {φ i } K i = {φ i } T [K] {φ i } C i f i (t) = {φ i } T {F(t)} Modal damping: ξ i = 2 ω i M i The response is obtained by applying convolution integral: p i (t) = h i (t) * f i (t) = 0 t f i (τ) h i (t œτ) dτ The solution in time domain is also called Duhamel s integral. The initial conditions are that speed and displacement are zero. The load is then {F i e }= [K] {φ i } = ω i 2 [M] {φ i } And the generalized bending moment is M i M i (x)=ω i 2 And the resulting normal stress σ D = N Σ i= 1 p i 0 x (x œs) m(s) φ i (s) ds M i (x) Z(x) where the unit response function is: h i (t œτ) = 1 sin ω M i ω id (t œτ) exp[œξ i ω id (t œτ)] 0 < ξ i < 1 id 2 ω id = ω i 1 œξ i The approach works for any wake!

9 Wake Types Machinery induced vibrations Foundation of the engine Axel line Depends on revolutions of the engine Etc. Propeller induced vibrations Number of blades sets the frequency Loading on the propeller Cavitations Transmitted as pressure impulses on hull or through axel line Ice-induced on plating Wave-induced on plating and hull girder

10 Springing vs. Whipping When the length end eigenfrequency of short waves coinsides with that of hull girder lowest eigenmode (2-node), the phenomena is called springing (resonant) Ships with low natural frequency, i.e. low stiffness to mass ratio Calculations suggest that springing may also contribute to the extreme response for some ships, but springing vibrations are generally more important for fatigue, up to 50%. When transient load causes hull girder vibrations the phenomena is called whipping Slamming causes impact load Due to this load vibrations occur In some wave conditions a ship may experience slamming loads for almost every wave encounter and then these two phenomena occur at the same time If the damping is quite low, this gives rise to continuous hull girder vibrations of varying amplitude. This illustrates that there is not always a clear distinction between whipping and springing. 2-nodes 1,5 1 0, ,5 25-0,5-1 -1,5-2

11 Whipping Whipping is caused by impact type loading due to harsh weather conditions The key issue is the relative motion and speed between the ship and waves The impact load causes transient vibrations on the ship, which is called whipping (compare to springing) 1,5 1 0,5 0-0, ,5-2

12 Whipping Wave bending Moment + Impact Response the time evolution of the stresses caused by the vertical bending moment, following severe slamming event"

13 The Relative Motion between Ship and Wave The relative motion is defined as difference between Ship vertical motion w(x,t) Wave height ζ(x,t) The notation z ( x, t) = w( x, t) ζ ( x t) z, z(x,t)>t, the bottom is in the air possible slamming occurs z(x,t)<-f, the deck gets submerged shipping of green water occurs For slamming to conditions need to be fulfilled The bow has to be in the air, z(x,t)>t The vertical speed of bow or stern has to exceed certain tresshold value w(x, t) F ζ(x, t) CG x T y

14 Total Time Derivative E ( X, Y, Z) is fixed ground coordinate system O (x,y,z) is moving coordinate system X E = U t + x where U is ship speed Z dx dt = U Y Total time derivative E U t z y D dx = + = U. dt t x dt t x O x X

15 F The Force Acting on Ship Frame Inertia component F I = D dt buoyancy dx m m ( mz! ) = ( mz! ) + ( mz! ) = mz!! + z! Uz!, t x dt t x F = ρg( T + z) B( z). The total force F = F + F I m z m 2 m since z! = z! = z!, we obtain ( x t) = z!, 2 t ρgδar, m + ρgδar, z t z z! < 0 z! > 0 z kvv m( z+ d z ) Iskuvoima [N] Bow separates from water (emmersing) Bow enters water (immersing) m( z) z B ( z+ d z ) z z+ d z aika [s] T laskettu mitattu β

16 Influence of Bow Shape Bad design Good design

17 Summary of the Process The total response is obtained by summing the impact and normal wave bending responses y(x, t) = N Σ i= 1 M Σ φ i (x) p i (t) + a k [ H h (ω k ) œx H θ (ω k )]sin(ω k t + α k ) k = 1 Strip-method for Hull Girder Response Creation of the Time History for Relative Motion and Speed Calculation of the Added Mass for Different Draughts M is the number of regular waves, a k is the amplitude of wave component and H h and H q are the response functions of heave and pitch Eigenmodes and -frequencies Calculation of Slamming Loads Calculation of Stresses

18 Rauma-Class L OA = 48.0 m B = 8.00 m Displacement 215 tons T = about 2 m v = 30+ knots.

19 The Eigenmodes 3D-FEM Usually beam model is sufficient for whipping calculations since we need only the hull girder bending modes (σ 1 -level) In present case 3D-FEM was used since Significant influence of superstructure Discontinuities Bulkheads The mass of water was included (added mass) by FEM analysis (infinite and finite size) Only the most significant modes are presented The modes with large displacements at bow are important since then the force works (W=F*u) Two first modes are beam modes while the 5 th mode includes also the superstructure deformations

20 Whipping-Analysis Changes in bow halved the load Normal stress presented in the figure below shows significant increase in the sagging moment

21 Vibration Tests Transducers can be used to get the time history of accelerations speed displacement Exciter test Done on board the ship to validate calculation models Unbalanced rotating mass is located somewhere on the ship The water depth should be at least 5x draught of the ship Masses should reflect actual conditions The aim is to detect resonance frequencies Sea trials Measurements on selected locations of the ship Engines running and wave-induced loads are present Criteria Stress levels and fatigue on structures Functionality of machinery and equipment Comfort of passengers, i.e. amplitude, duration and frequency of motion

22 Eigenfrequencies for Beams

23 Eigenfrequencies for Rectangular Plates

24 Summary Vibrations can be harmful for people and equipment Many sources: Waves (<1Hz) Machinery (600RPM=10Hz) Propeller (140RPM, 4 blades ) We should omit resonance, starting point is to ignore the frequencies of wake Beam and plate theory can be used to assess the eigenmodes