Part C
Calculation methods and experimental investigations
P.A. Kolkman
T.H.G. Jongeling
The three manuscripts (parts A, B and C) were put in book form by Rijkswaterstaat in Dutch in a limited edition and distributed within Rijkswaterstaat and Delft Hydraulics. Dutch title of that edition is: Dynamisch gedrag van Waterbouwkundige Constructies.
Ten years after this Dutch version Delft Hydraulics decided to translate the books into English thus making these available for English speaking colleagues as well. Because of in the text often referred is to Delft Hydraulic reports made for clients (so with restrictions for others to look at) we decided to limit the circulation of the English version as well. However, the books are of value also without perusal of these reports.
The task was carried out by Mr. R.J. de Jong of Delft Hydraulics. Translation services were provided by Veritaal (www.veritaal.nl).
Delft Hydraulics 2007
Table of contents Part C List of symbols Part C
1 INTRODUCTION...1
2 ANALYSIS OF DESIGN AND CONDITIONS ...5
3 CALCULATION METHODS FOR DYNAMIC BEHAVIOUR OF STRUCTURES IN WATER...9
3.1 Considerations for calculating in the frequency domain or in the time domain 9 3.2 Calculation of the added water mass for two-dimensional situations (ignoring wave radiation) ...12
3.2.1 Assessment of the frequency range in which the added water mass is not frequency-independent anymore ...12
3.2.2 Added water mass calculated with potential theory ...13
3.2.3 Assessment of added water mass based on schematized flow pattern ...14
3.2.4 Examples of complete calculations ...15
3.3 Calculations in the frequency domain ...20
3.3.1 The single (degree of freedom) mass spring system ...21
3.3.2 The single (degree of freedom) mass spring system in water ...24
3.3.3 Response of a double (degree of freedom) mass spring system (direct method) ...26
3.3.4 Modal Analysis in case of a double (degree of freedom) mass spring system.28 3.3.5 General formulation of a system with multiple degrees of freedom ...30
3.3.6 Coupled systems with structure components and fluid components...34
3.3.7 Examples of coupled systems with structure components and fluid components ...35
3.4 Calculations in the time domain using the indirect method ...43
3.4.1 Modal Analysis and impulse-response method in case added mass and damping are frequency-independent ...43
3.4.2 The impulse-response function in case added mass and damping are frequency- dependent...44
3.5 Calculations in the time domain with the direct method...48
3.5.1 General ...48
3.5.2 Response of a single mass spring system to an external load ...48
3.5.3 Coupled systems with structure components and fluid components...51
4 CALCULATION METHODS FOR WAVE IMPACTS ...61
4.1 General ...61
4.2 Impulse theory ...62
4.3 The linear shock wave model...64
4.3.1 Wave impact against a rigid wall ...64
4.3.2 Rigid wall and air-water mixture...65
4.3.3 Wave impact against a compressible wall...66
4.4 The non-linear shock wave model...67
4.5 The flow-pressure model (ventilated shocks)...70
4.6 The air compression model...71
4.6.1 The linear air compression model ...72
4.6.2 The non-linear air compression model ...73
4.7 Numeric calculation of the pressure function (in time) in case of a wave impact ...76
4.9 Influences on impact load due to a responding structure ... 79
5 SCALE MODELS ... 81
5.1 Introduction... 81
5.1.1 General... 81
5.1.2 Investigation strategy for a real project... 82
5.2 Scale rules and scale effects in case of vibration investigations and investigations of wave loads ... 83
5.3 Classification of scale models for vibration and wave impact investigations... 90
5.3.1 Classification with respect to reproduction of geometry ... 90
5.3.2 Classification with respect to reproduction of dynamic properties... 92
5.4 Possible critical aspects of dynamic models... 94
5.5 Verification of the model technology... 95
5.6 Measuring system and data processing... 101
5.6.1 General... 101
5.6.2 Instrumentation ... 102
5.6.3 Monitoring and registration of measuring signals ... 103
5.7 Elaboration of measuring results... 104
5.7.1 General... 104
5.7.2 Statistical elaborations ... 105
5.7.3 Elaborations in the time domain ... 107
5.7.4 Elaborations in the frequency domain ... 110
6 EXAMPLES OF SCALE MODELS FOR DYNAMIC INVESTIGATIONS ... 119
6.1 Rigid model with flowing water, for vibration investigations... 120
6.2 Rigid model for investigation of wave load... 123
6.3 Single (degree of freedom) mass spring system for vibration investigations; translatory... 130
6.4 Single (degree of freedom) mass spring system for vibration investigations; rotating... 138
6.5 System with multiple degrees of freedom in case of floating gate ... 141
6.6 Multiple (degree of freedom) mass spring system for response investigations in case of flow and waves ... 145
6.7 Continuous-elastic model for vibration investigations ... 148
6.8 Continuous-elastic model for investigations of wave loads and vibrations .... 152
7 INVESTIGATIONS OF PROTOTYPE STRUCTURES... 159
7.1 General... 159
7.2 Vibration measurements ... 160
7.3 Wave impact investigations... 163
7.4 Elaboration of measuring results... 164
7.5 Experiences with respect to vibration and wave impact measurements... 164
8 REFERENCES... 171
8.1 Delft Hydraulics Reports (in Dutch) ... 171
8.2 Other reference material... 177
APPENDIX I ... 181
DEDUCTION OF THE RESPONSE FUNCTION IN THE TIME DOMAIN FROM THE TRANSFER FUNCTION IN THE FREQUENCY DOMAIN... APPENDIX II ... 187
SCALE RULES AND SCALE EFFECTS FOR INVESTIGATIONS OF DYNAMIC BEHAVIOUR OF SCALE MODELS... APPENDIX III... 199
DESCRIPTION OF A CALCULATION PROGRAM FOR THE DETERMINATION OF THE ADDED WATER MASS FOR A FLAT ROD WITH RECTANGULAR SECTION (STRIP) IN WIDE WATER IN CASE OF TRANSLATORY AND ROTATING
VIBRATIONS... INDEX BY SUBJECT (PART C)...207 Biographies of the authors...212
List of symbols Part C
a = mean distance between water front and structure (m) ai = amplitude factor of the ith natural vector (1, ….i, …..n) (m) an = amplitude of the nthe vibration peak (m)
A = surface area (m2)
A = flow area beneath the gate (m2) Ap = cross-section area of tube (m2) B = beam of ship (m)
c = damping constant (Ns/m)
cc = celerity compression wave in the structure (m/s)
cw = damping due to water, also called added damping (Ns/m) cw = celerity compression wave in water (m/s)
C = damping matrix (Ns/m)
Ca1 = Cauchy number related to the spring stiffness Ca1 = k/(ρv L2 ) Ca2 = Cauchy number related to the module of elasticity Ca2 = E/(ρV2) CA = discharge coefficient related to the opening of the gate (-)
d = initial thickness of enclosed air cushion (m) D = draught of ship (m)
D = diameter of tube (m)
Da = damping number = c/(ρVL2)
ei = ith natural vector (n=1, …..i, ….n)
f = excitation frequency of the flowing fluid (1/s)
f = vector of the force acting on the nods of the finite elements (N) F = force (or load) (N)
F = force vector (N)
Fr = Froude number = V/ gh or V / gL Fw = load caused by flowing fluid (N)
Fw = amplitude of a periodic load caused by flowing fluid (N) F0 = amplitude of force (N)
Fo = floating force (N)
g = gravitational acceleration (m/s2) G = weight (N)
h = water depth or depth of structure under water (m) hc = culvert height (m)
h1 = upstream water depth h2 = downstream water depth
H(f) = transfer function in the frequency domain i = index indicating the ith Eigenvector I = impulse (vector) (Ns)
k = spring constant (N/m) k = slamming coefficient
Kl = compression module of air (m2/N) Kw = compression module of water (m2/N) K = stiffness matrix (N/m)
kw = hydrodynamic stiffness, or added water stiffness (N/m) L = length, in particular length of tube (m)
Lw = representative length of added water mass (m) LC = perimeter of the tube cross-section (m)
Lculvert = length of culvert (m) L1 = length of a tank (m)
L2 = length of connecting culvert between tank and outer water (m)
m = mass (kg)
mw = added water mass (kg) M = mass matrix (kg) Ma = mass number = m/(ρL3)
n = n-direction is perpendicular to a surface n = number of cycles of natural oscillations
n = scale factor; an index refers to the quantity involved N = a number, indicating distance in N LΔ
p = pressure (N/m2) p0 = initial pressure (N/m2)
q = discharge per unit width (m2/s)
ql = discharge of left gate per unit width (m2/s) qr = discharge of right gate per unit width (m2/s) Q = discharge (m3/s)
Q0 = permanent part of discharge (m3/s) Q' = time dependent part of discharge (m3/s)
r = complex number (real part is related to damping, the imaginary part is related to the frequency) (radials/s)
R = radius, half of internal tube diameter (m) Re = Reynolds number = vD/ν (-)
s = damping term for a damping force which is proportional to the square of the velocity
of the vibration movement (Ns2/m2) S = stiffness matrix (N/m)
S = Strouhal number = fL/V (-)
t = time (s)
T = duration (s)
T = vibration period (s)
u = velocity component in direction x (m/s) v = velocity component in direction y (m/s) v = velocity vector (m/s)
v0 = velocity of vibrating object (m/s) v0 = initial water velocity (m/s) V = volume (m3)
V = (reference) fluid velocity (m/s)
V' = time depending part of fluid velocity (m/s) Vs = ship velocity (m/s)
w = vector of (linear independent) displacements of nodes (m) We = Weber number = 2/
wLV
ρ σ
x = coordinate in horizontal direction x = displacement in direction x
ˆx = amplitude of vibration in direction x
y = coordinate in horizontal direction (perpendicular to x direction) y = displacement in direction y
Y = amplitude of vibration of object (m) Y = displacement vector (m)
ˆY = amplitude of harmonic movement y(t) (m) z = coordinate in vertical direction
z = water surface variation (m)
z = dy/dt (m/s) or water surface variation in case of oscillation (m) α = ratio of potential Ф and velocity vstructure (m)
α = ratio between increase of gate discharge (per unit width) and water surface variation (m/s)
α = phase angle (radials) α = air percentage
α* = air percentage after the front of the shock wave
γ = relative damping
γ = Poisson constant
δ = gap height (m)
δ0 = time independent part of gap height (m) ΔE = energy loss or energy transfer per period (Nm)
Δh = difference in water depth between upstream and downstream (m) ΔH = hydraulic head difference (index refers to over what location) (m) ΔH' = time dependent part of hydraulic head difference (m)
ΔL = mesh distance (m)
Δp = pressure difference (N/m2) Δt = time step (s)
Δq = discharge per length unit L and per unit width (m/s) ΔΦ = step in potential (m2/s)
є = ratio between wall velocity and velocity of incoming water (m/s) Φ = velocity potential (m2/s)
θi = phase angle of the ith natural frequency (1…I, ….n) (rad) ψ = root-mean-square (RMS) value of displacement (m) ρ = density of a fluid (kg/m3)
ρc = density of structure material (kg/m3)
μ = discharge coefficient or contraction coefficient μx = mean value of displacement in direction x (m) υ = kinematic viscosity (m2/s)
ξ = loss coefficient (index refers to what related) σ = surface tension (N/m)
σx = RMS value of displacement in direction x (m) τ = impact duration (s)
τ = time displacement (s)
ω = angular frequency of the vibration (rad/s)
if = ith natural frequency of the vibration (1….i, ….n) (rad/s) ωn = natural (angular) frequency of the vibration
1 INTRODUCTION
Part C of this manual offers an overview of the calculation methods and experimental investigations for the benefit of designing new structures in a responsible way, improving existing structures and deepening our knowledge of dynamic phenomena. The investigations themselves will often require the input of specialists. This part aims to provide the designer with some understanding of the available methods of investigation.
Two warnings in advance:
• This part goes into matters more deeply than the first two parts and indicates limitations; it is therefore more concerned with background information.
• Although the reader may occasionally get the impression that everything is known, the available methods of investigation are far from perfect. Many data are lacking and each approach has its limitations.
The methods of investigation that are available to test a design are: a. Analysis and desk research.
b. Carrying out calculations in the frequency domain or in the time domain. c. Using scale models.
d. Carrying out measurements of existing structures.
This order may be representative of the operating procedure in case of an analysis of a design for a structure that will be built, provided that method c, due to the costs and the duration of the investigation, will often only be used when methods a and b indicate the necessity for it. If the dynamic behaviour cannot be forecasted accurately, then method c will qualify earlier (both in time and in importance). Methods b and c will run simultaneously: the scale model serves to discover phenomena, and the calculation model will be started up according to circumstances, and will be calibrated with the available measuring results. Next, the calculation model will be elaborated with a number of variants under different conditions, and finally the selected variant will be verified again in the scale model.
The order of methods a through d mentioned above is also followed in the chapter arrangement of this Part C.
ad a.
In order to carry out an analysis, naturally existing knowledge must be mobilized. Therefore a specialized descriptive bibliography is available, see in Delft Hydraulics A110. Flow-induced vibrations naturally do not only occur at hydraulic structures. Vibrations induced by flowing air or fluid (of wings, plating and other components, propellers, flow machines, flap gates etc.) may also occur in maritime engineering, aeronautical engineering and mechanical engineering.
Analysis should preferably be carried out based on criteria that relate to the design, stiffness and damping. Using a checklist may serve as a starting point; for possible causes of vibrations, see for example Part A, Chapter 1. Refinements may be possible by checking the design on the basis of one or more criteria (for example as mentioned in Part A, Paragraph 4.4.6).
The evaluation of the design with respect to the occurrence of wave impacts is
especially an evaluation of the shape of the design. Only when it is not possible to change the shape in such a way that these impacts may be avoided, it will be necessary to assess or calculate the effect on the structure.
This part of the manual deals with the structural aspects of the design and the vibration and wave impact phenomena; most aspects with respect to vibrations and wave impacts have already been dealt with in Part A and Part B.
ad b.
As regards vibrations, Chapters 2 and 4 of part A have instigated calculating in the frequency domain. Chapter 3 of this part will discuss these matters more extensively. Calculating in the frequency domain is not useful for wave impacts. Calculating in the time domain is also introduced as a method in Chapter 3 of this part as well.
Apparently, often too many basic data are unavailable in order to make a complete calculation useful. If the load is assumed to be known, it is however possible to calculate the response of the structure. For certain kinds of vibrations of gates and valves, the extent of self-excitation may be determined on the basis of calculations.
Chapter 4 discusses various calculation methods for the determination of wave impacts in greater detail.
ad c.
There are many kinds of scale models that are used for the investigation of vibrations or response in case of wave impacts (overview or detailed models, completely rigid or
completely elastic models, and everything in between). In some cases models in wind tunnels may be used as well. Each model offers possibilities of investigation, but also has its
limitations. Scale models may have important scale effects. Scale models may be a very important tool in the design process for the investigation of dynamic phenomena. On the basis of the current state of affairs, scale models are in certain ways even more important than calculation models, as they may sometimes reveal unexpected phenomena. Often calculation procedures may only be set up on the basis of scale model observations. The calculation model serves to elaborate quickly with many variants in case of a systematic variation of conditions and design parameters. Chapter 5 provides an overview of the possibilities of scale models. Chapter 6 provides examples.
ad d.
Measurements of existing structures do not occur often, as in reality structures usually suffice reasonably well. In those cases there is no need regarding the structure itself to carry out more detailed measurements. But there is a number of structures that do not meet the requirements, giving cause for repairs and extra maintenance. Moreover, the supervisor needs to be continuously aware of the fact that vibrations or wave impacts may occur. In general, measurements of existing structures are indispensable, first of all as a means of verifying the calculation and scale model investigations that are used and secondly to put into perspective what might be accomplished with these methods. Practical experiences and measurements also give cause for revising the boundary conditions or the theoretical concept. Examples of this are:
• observing cavitation and the ensuing occurrence of great dynamic loads; • the growth of mussels and other kinds of fouling on structure components; • the suction of air;
• damages of a gate edge, as a result of which vibrations are generated;
• the being out of action of gates, as a result of which the conditions at the other gates are more severe;
• the occurrence of high-frequency plate vibrations; these can never be forecasted adequately on the basis of scale model investigations in the design phase.
Prototype measurements are the final piece of the reviews of model and scale effects that are always present in scale models. Chapter 7 discusses a number of special aspects regarding the measurement of prototype structures.
The following results are shown as examples of an interesting measurement in
prototype that was carried out in January, 1995, i.e. the vibration measurements carried out at the gate of the tide lock at Ravenswaay. The results are rather surprising and as yet there is no definitive explanation for the cause of these vibrations.
Figure C1.1:
Transverse waves upstream of the vertical lift gate in the tide lock at Ravenswaay, generated by vibrations of the ‘closed’ gate.
Figure C1.1 shows transverse waves that relate to the vibrations of the gate (80 m span) in the situation of the ‘closed gate’. The gate is of the same type as the one that is used in the storm surge barrier at Krimpen (Part A, Chapter 6, Example 6.2b), though in case of the tide lock at Ravenswaay a rubber bottom edge has been used to obtain a good seal. The vibrations of 3 Hz were horizontal and vertical combined, with the greatest acceleration
horizontal in the flow direction: 3 m/s2 (deflection approximately 10 mm). The waves are subharmonic standing transverse waves (see for a more detailed description of this type of transverse waves Part A, Paragraph 4.6) that could be observed on both sides of the gate.
The fact that vibrations occurred when the gate was completely closed was surprising in itself, and it was only possible in case there was a leakage gap. The leakage was probably due to causes relating to the closing procedure or due to fouling on the sill. The vibrations were generated, starting from a hydraulic head of 2.5 m.
2 ANALYSIS OF DESIGN AND CONDITIONS
As Part C especially focuses on calculation methods and experimental investigations, only little attention is given to the general analysis of the design.
Each design begins with a global analysis in which the expected wave and flow conditions are discussed and on the basis of which the structure may be determined more specifically and in greater detail.
Preceding a detailed study and/or investigation of the dynamic behaviour of structure components that may be critical (such as gates and trash racks), first the local boundary conditions for those components need to be determined separately. Based on the hydraulic head of the structure, the local hydraulic head of the gate and the flow velocity at a trash rack need to be deduced. The local wave conditions need to be determined on the basis of the global wave conditions. For example, a different combination of water level of the hydraulic head applies to each opening percentage at a gate or valve. The wave height may vary strongly with the water level and will also be influenced by the discharge that flows through the structure.
The requirement that a structure should not experience great dynamic loads, may lead to limitations in the operational management of the engineering structure.
These points will be further discussed below, in which the aspects of vibrations and wave impacts will be treated separately.
Regarding the local hydraulic head and the local wave conditions
This especially focuses on the local hydraulic head, for as far as local wave conditions are concerned, in any event these will require a calculation using a standard calculation program. The local waves may be considerably higher and steeper than the incoming wave due to reflection, canal narrowing, sloping (ascending) bottom and local shapes that function like a hoop net.
At a gate or a valve, the local hydraulic head may deviate considerably from the hydraulic head of the entire structure. As a result of this, the determination of the local hydraulic head first requires an impression of whether the inertia of the water in the culvert should be included in the calculation or not. If this is the case, the calculation of the local hydraulic head becomes very complex (see Delft Hydraulics Report R1506). The following procedure is used.
• First, the gate drag term, ξgate, or the coefficient of the flow, Ca, needs to be determined for each opening percentage. This is defined as:
2 2 culvert gate gate V H g ξ Δ = (C2.1)
(in which ΔH = hydraulic head, V = flow velocity, g = gravitational acceleration) and: 2
a gate gate
Q C A= g HΔ (C2.2)
(with Q = discharge, Agate = flow-through opening of the gate)
• Given the closing program of the gate or valve, the drag term needs to be determined as a function of time.
• Next, the movement equation of the water in the culvert needs to be analyzed. This equation is as follows:
( ) 2 d
2 d
culvert culvert culvert drag inertia culvert gate
V L V
H H H
g g t
ξ ξ
Δ = Δ + Δ = + + (C2.3)
• By first assessing how the discharge is distributed over time when the last term (the inertia term) in Equation C2.3 is being ignored, it is possible to obtain a first
assessment of the discharge over time. Consequently, a first estimate may be made of the magnitude of the inertia term, and whether this is important relative to the total hydraulic head. The term ‘apparent hydraulic head’ is introduced:
d d culvert culvert apparent L V H H g t Δ = Δ − (C2.4)
• As the term dVculvert/dt is only negative during the closing of the gate, the apparent hydraulic head of the structure during closure is greater than the real hydraulic head. For this reason it may be advisable to select a slower movement velocity for the closing of a gate or valve, than for the opening.
• If the apparent hydraulic head that is calculated in this way is not much greater than the real hydraulic head (5% seems to be a reasonable limit in this case), then a dynamic calculation of the closing program may not be needed.
• If a dynamic calculation is chosen, this means a calculation in the time domain with a small time interval, Δt: (using Equation C2.3) each time interval is calculated as follows: d d culvert culvert V V t t Δ = Δ (C2.5)
The result may be translated into a discharge as a function of the gate position.
• For the first or the last trajectory in which the valve is nearly or completely closed, the gate drag coefficient ζgate becomes very great, even up to infinitely great. To
circumnavigate this problem in the calculation, a constant discharge coefficient, Ca (related to the gate opening) is used for the last part of the closing program, as well as the assumption of a constant hydraulic head, a negligible resistance of the culvert and an increasing or decreasing gate opening linear with time. This results in an analytical solution.
The dynamic equation for the discharge then looks like this:
2 ( )
a gate inertia
Q C A= g HΔ − ΔH (C2.6)
in which it may be proven by way of an indirect demonstration that if Agate changes linear with time, the discharge increases or decreases linear with time as well, in which the inertia hydraulic head becomes a constant. From Equation C2.6 it may be deduced that:
d( ) d d 2 ( ) d d d a gate culvert culvert C A L Q Q g H t = t Δ − gA t (C2.7)
As all coefficients with time are constants, dQ/dt follows from the solution of a quadratic equation.
• From the calculation of the discharge history mentioned above, the local pressure differences across the gate may be deduced.
• If necessary, the absolute pressure is determined as well; this may be important in connection with the occurrence (or not) of air suction or cavitation. To be able to calculate better the local pressure at the gate or valve, the inertia of the water in the culvert is subdivided into inertia upstream and downstream of the valve. It also needs to be taken into account, that directly downstream of the gate, the pressure is extra low (relative to the pressure a little bit further down in the culvert). This relates to the contracted nappe that may give cause for the recovery of potential energy.
• In case of a shipping lock, the lock chamber is already partially filled before the gates of the filling culvert are fully open. This means that for each gate position there is a different normative external hydraulic head of the filling or emptying culvert. If the lock chamber is filled through multiple gates, then the situation with one or more gates out of action needs to be considered as well. For it is possible that the selection of the complete filling program strongly relates to the avoidance of vibrations, cavitation and air suction.
Regarding vibrations
• It is useful to assess the resonance frequencies as a function of the span of the structure; this may certainly influence the choice of the span.
• From the data concerning the Strouhal number (e.g. Part A, Paragraphs 4.2, 5.2, 5.3, 5.4 and 5.7) an impression may be obtained about the dominant excitation frequencies due to turbulence in the wake of the structure and of structure components.
• It may be verified whether the design of a gate or valve may give cause for bath plug vibrations or for other vibration modes with self-excitation. About this, see Part A, Chapter 7 to begin with.
• Sliding gates experience a greater damping in case of possible vibrations and are safer where vibrations are concerned.
• In case of gates and valves the choice of the type of seal (or the omission of these) is very important for the possible occurrence of vibrations. For this, see also Part A, Chapter 7.
• In case of gates, not only the flow through the gate opening plays a role, but also the remaining surrounding flow. The surrounding flow and the pressure distribution in the gate shaft are important for the gate load and the possible generation of vibrations. • For a (storm surge) barrier in a tidal area it makes a lot of difference whether it is
eventually decided that the barrier is closed around the time of the turning point of the tide or that it is closed after waiting until a critical situation is reached. The reason for the decision to close the gate around the time of the turning point may be the wish to avoid powerful forces during the movement of the gates, or to avoid the probability of dynamic behaviour or the limiting of translatory waves.
Regarding wave impacts
In the following (and this essentially applies to the whole of Part C), periodic wave loads are not further discussed, as there is no interaction with the dynamic properties of the structure.
Wave impacts may occur in those situations in which the wave (for example when it is reflected) is locked in and in situations in which the moving water surface hits a wall that runs parallel to this water surface. Even a water surface that slowly moves up and down
horizontally may generate a wave impact, if it hits a horizontal ceiling. In the same way as discussed in Part B, there is a number of aspects that need to be taken into account when designing a structure.
• In connection with the avoidance of wave impacts, the most important aspect is the positioning of the structure relative to the waves that are present.
• In those cases in which waves hit the structure, the water needs to be capable of moving horizontally or vertically unobstructed. As horizontal wave propagation is not possible in case of a closed gate, vertical parts of the structure preferably need to be smooth; the presence of, for example, horizontal ceilings and closed working
platforms in the wave zone must be avoided. Wave reflection due to a wall or a closed gate may be observable even from great distances; therefore a bridge deck at a certain distance from the wall may be in a danger zone with respect to the occurrence of a wave impact.
• Horizontal girders of a gate or gates that are located at a critical height, are in less danger of being loaded by wave impacts during the outflow of a discharge, as it is much more difficult for waves to reach the gate in case of flow.
• If a horizontal element (e.g. a platform or a horizontal girder) is unavoidable, it must be perforated and preferably be replaced by a lattice structure with rods that are sharp or rounded at the bottom, instead of flat.
• If waves are capable of penetrating into a culvert in which the gate is positioned, then it is advisable to design the gate shaft in such a way, that a connection is maintained between the shaft and the culvert at the side of the incoming waves. This prevents the wave from being locked in and the water from being slowed down too suddenly. • To prevent ‘rattling’ due to wave impacts in case of a barrier that is surrounded on
both sides by the same water level, prestressed sliding supports may be used.
The evaluation of the design requires specialized knowledge. Part A (flow-induced vibrations) and Part B (wave impacts) contain much of that knowledge in summary.
As far as the possible sensitivity to flow-induced vibrations is concerned, we refer to Part A, Chapter 1, in which a classification of possible vibrations may be found, Chapter 4, which deals with various kinds of vibration mechanisms at gates, Chapter 5, which deals with flow-surrounded objects and Chapter 7, which discusses remedies.
The evaluation regarding wave impacts is especially an evaluation of the design. If the design cannot be adapted in such a way that wave impacts may be avoided, the effect of an impulse load must be assessed more closely. An analysis of the dynamic properties of the structure is then required. About this, see Chapter 6 of Part B.
3 CALCULATION METHODS FOR DYNAMIC BEHAVIOUR OF
STRUCTURES IN WATER
Calculations of the dynamic behaviour of structures in water are not carried out very often. On the one hand this is to do with the complexity of the calculations themselves and on the other hand it is to do with the many unknown factors and insecurity concerning the
magnitudes that need to be introduced. Also, the mathematical-physical description of the interaction between fluid and structure is not always that easy. This means that the
development of knowledge in the past has mainly been focused on the avoidance of dynamic phenomena. The possibilities of calculation methods however are increasing rapidly. Next to this, a lot of understanding and experience has become available thanks to all the effort spent on large-scale structures. This especially applies to discharge sluices and storm surge barriers in the delta area of the Netherlands. This chapter therefore discusses at length the various available calculation methods.
3.1 Considerations for calculating in the frequency domain or in
the time domain
There are various methods to calculate the dynamic behaviour of ‘dry’ structures in case of a given load. Not all methods are suitable for structures in water.
First of all, there is always a need to find fundamental equations that accurately describe the problem. In case of vibration problems, inertia always plays a role, both of the vibrating masses and of certain fluid elements. By taking periodic solutions as a starting point, in which all terms include a sinus function or a complex e-power, the acceleration is replaced by a displacement, multiplied with -ω2, after which the sinus or e-power may be removed from all terms. Only when this direct approach does not yield sufficient results, it will
probably be decided to change to a more complex calculation in the frequency domain or to a calculation based on a mathematical model in the time domain.
Whether a calculation in the time domain may lead to results, depends on the question whether the extra mass, damping and stiffness terms that are generated by the water, depend on the frequency of the response or not. For situations in which the added water mass is frequency-dependent, the natural frequencies of a structure in water cannot be calculated directly. The fact of the matter is, that for this calculation the added masses need to be known, and these again depend on the natural frequency. Calculation is only possible through
repetition. By assessing the natural frequency, the frequency-dependent added water mass (or the matrix expression for it) may be calculated; by using this, the calculation of the natural frequencies may be carried out again.
The frequency-dependency relates to the radiation of waves that are generated by the vibrating or moving structure, and therefore to the presence of a free water surface. If the vibration is high-frequency or the movement is of short duration, then the radiated waves have a short wave length and the dynamic pressures that relate to the waves may only be observed across a limited part of the depth (relative to the free water surface). The influence of the wave radiation therefore disappears at higher frequencies. See further Paragraph 3.2.1.
a. Direct calculation in the frequency domain; this is especially adequate in case of periodic loads. As the frequency is known, there is no fundamental difficulty when the added water mass is frequency-dependent.
b. Indirect calculation in the frequency domain and in the time domain, while using the Modal Analysis (the theory of eigenvalues). Random loads may be considered; using this method however is limited to those conditions in which the added water mass, damping and stiffness are not frequency-dependent.
c. Indirect calculation in the time domain with the impulse-response function technique, especially adequate for calculation of the response to loads of short duration. With this method it is also possible to include frequency-dependent terms in the calculation. The method has been used in the calculation of forces due to the collision of ships against, as an example, fenders, fender walls, mooring posts; instead of an independent
external load, the load was generated due to the compression of, for example, one or more fenders.
d. Direct calculation in the time domain, adequate in case of impulse loads, but only applicable for situations in which the added water mass, damping and stiffness are frequency-independent.
ad a: In case of a linear system it is possible to calculate directly in the frequency domain, as a linear system in case of a harmonic load also generates a harmonic response with the same period. Consequently, time-dependent factors (such as sin(ωt) or eiωt) may be removed from the equations by division, as a result of which the remaining equations may be solved directly. This applies both to a single and to a multiple system. ω = angular frequency, i = indication of an imaginary number, t = time. If there are damping terms (i.e. proportional to the vibration velocity), then terms with both sin(ωt) and cos(ωt) are formed. This means that two systems of equations are formed, from which the phase shift between load and response may be determined. This outcome may also be obtained through complex calculations. If the periodic load F(t) is written in terms of Feiωt, then the movement y(t) of each mass point may be written as or Ŷeiωt. The eiωt may be removed from all terms by division. This way of
differentiating with time is the same as multiplying with iω. If ˆF is a real number, then the outcome for Ŷ consists of a real and an imaginary part, which results in the phase shift between the harmonic force and the harmonic vibration movement. Here again, two systems of equations are formed, i.e. by equalization of all real terms and all imaginary terms respectively. Here, the two equations directly lead to a solution; this solution is referred to as the particular solution.
As the superposition principle applies to linear equations, the general solution, i.e. that of a freely damping out vibration, may be added to the particular solution, thus
complying with the initial conditions at the moment in time t = 0. Calculation of the free vibration however is difficult when the added water mass is frequency-dependent. Paragraph 3.2.1 discusses this in more detail, using an example.
The direct method of solution is adequate in case a response diagram needs to be determined (Figure A2.3 in Part A).
ad b: Modal Analysis is used in the frequency domain (response in case of harmonic load), but also in the time domain and it is applicable to systems with multiple degrees of freedom with a load that is distributed randomly over time. Paragraph 3.3.5 discusses this method in more detail. Here it suffices to give a brief summary of the various elements of this calculation method.
The response of the structure is considered to be factorized into components,
eigenvalues or ‘modes’. Each mode has one eigen (or natural) frequency, one eigen (or natural) vibration mode and one eigne value for the load; these are univocally coupled to each other. The amplitude relative to the corresponding mode may take on different values. Each of the vibration modes corresponds to one of the free vibrations
(therefore without external load) of the undamped system. There is an eigen value for the load that corresponds to each mode (in fact, the load distribution) that statically generates the same deflection as the one that relates to the corresponding vibration mode of the free vibration. The amplitude of the free vibration and that of the
corresponding load initially are not fixed. It may be shown that this load distribution, regardless of the frequency in which this load makes contact, is always accompanied by the same vibration mode and that the response curve in the frequency domain is identical to that of a single mass spring system. By factorizing the actually present load into the corresponding eigenvalues of loads (each therefore with the right load distribution across the masses as well), the amplitude of each of the eigenvalues of the loads is now fixed. At this point it is possible to calculate the response in time for each eigenvalue of the load, in the same way as for a single mass spring system. The sum of these responses results in the eventual response of the structure to the load. In this calculation, the structure may also be damped, provided that the damping is equally distributed across the springs or across the masses, proportional to the magnitude of the spring stiffness or the mass concerned.
ad c: Using the impulse-response function is a method for calculating the response to a randomly distributed load in case the response of the structure to a load of short duration (the so-called unit impulse) is known. This load may be a known external load or it may have been generated by impressions of an elastic element (for example during a collision); also, in case the spring is non-linear, it is possible to calculate the response. For structures in water, in which the added damping and mass may be frequency-dependent, it is of course not possible to determine an impulse response. For an object that is not deformable however, and that moves in the water, a solution has been found. Paragraphs 3.4.1 and 3.4.2 discuss in what way the response to the unit impulse may be determined.
ad d: Calculating directly in the time domain means that the calculation occurs in small steps. The second law of Newton (F = m*a) applies to each mass. The force F that operates on the mass m and causes an acceleration a, consists of the external load, of spring forces and damping forces. In case of complex systems, calculating with this method requires a lot of time and in practice it is therefore less suitable for the determination of the response to a periodic load for an entire frequency range. One advantage is that in case of calculating directly in the time domain, non-linear components may be included.
All calculation methods mentioned above relate to structures with an external load or a freely moving or vibrating structure (for example, after an impact has occurred). The structure is schematized into a system consisting of masses, springs and dampers. The water generates both the external load and the passive forces, such as the added water mass, water damping and spring stiffness. As the water itself may also include components with their own natural vibration period (in case of fluids these are referred to as oscillations, rather than vibrations), in some cases it may be better to describe the fluid together with the structure, as one system
with, naturally, multiple degrees of freedom. Examples of such systems have been included in this chapter (Paragraphs 3.3.6 and 3.3.7).
3.2 Calculation of the added water mass for two-dimensional
situations (ignoring wave radiation)
3.2.1 Assessment of the frequency range in which the
added water mass is not frequency-independent
anymore
Wave radiation is generated when, in a situation with a free water surface, water is displaced in horizontal direction due to the vibration of the structure. If the wave length of the radiated wave relative to the water depth is small, then the influence of the wave radiation is small as well. In that case, the velocity at which the wave travels is small, so that it radiates little energy. The pressures in a vertical due to the radiated wave decrease exponentially, when the distance from the water level increases; this distance needs to correspond to the wave length. The influence of the wave therefore is also smaller when the wave length is smaller. This results in the fact that h/λ may be taken as a characteristic parameter of the influence of the wave radiation, and that again turns out to be proportional to ω2h/g.
In this, λ is the wave length, h is the water depth, or the depth location of the structure relative to the water level, and ω is the angular frequency of the vibrating structure. The vibration may be regular or may be a response vibration generated by an impact.
In order to investigate the influence of ω2h/g on the added water mass (and on the added damping as well), a study has been done (Delft Hydraulics Report W254) concerning a gate that is part of a vertical wall, and in which the gate is vibrating horizontally. The results may be found in Part A, Figure A3.1. If the water depth and/or the frequency are so great that ω2h/g > 10, then the CL value (a measure of the added water mass) appears not to experience any influence from the radiating wave. In case this influence is no longer present, then the CL value has also become independent of ω2h/g, and therefore of the vibration frequency. The water damping due to wave radiation may also be ignored at these ‘high’ frequencies. In the equation: if 2 10 h g ω > (C3.1) then the added water mass is frequency-independent and the water damping due to wave
radiation may be ignored.
Other investigations also seem to confirm this result (for example investigations of a ship oscillating in transverse direction, Fontijn (1975)). Theoretically it may be demonstrated that, also for very low values of ω2h/g, the added water mass tends toward a constant,
corresponding to the value found in case the water level is a rigid wall. For practical purposes this is not relevant, as dynamic phenomena no longer occur at these low frequencies.
In reality, the application of Equation C3.1 means that structures such as gates and trash racks are sufficiently stiff (i.e. showing a sufficiently high ω) for carrying out calculations with a constant water mass. In case of structures in a closed culvert, wave radiation plays no role, so that there too the added water mass is frequency-independent.
In case of long cylinders positioned in flowing fluid or in case of floating structures, the movement frequencies may be so low that there is frequency-dependency. Also the
periodic wave excitation is situated in that part of the frequency range in which the added water mass is strongly frequency-dependent. In calculating the collision forces during the mooring of ships at fender structures, an added water mass and frequency-dependent damping also need to be taken into account.
3.2.2 Added water mass calculated with potential theory
In Part A, Paragraph 3.2, the concept of added water mass has been discussed. For a vibrating structure in ‘drowned’ condition (structures that are below the water surface or in a culvert), the influence of wave radiation may be ignored. This also applies to structures that penetrate the water level, provided the vibration frequencies are high enough (see previous paragraph).
The influence of the free water surface is represented by the condition of constant pressure, in case the wave radiation is ignored. Contrary to the situation with a rigid wall, in case of a free water surface a flow component may also be present that is at right angles to the water level.
In case of still water, the flow that is related to the added water mass may be
calculated as potential flow. In that case, the following preconditions apply to the outlines of the fluid area:
• the velocity component at fixed walls, and at right angles to the wall, equals 0; • the precondition that the pressure is constant applies to the free water surface, which
translates into the precondition that the potential at that location equals 0;
• a vibrating wall is replaced by a fixed wall that is modelled with sinks and sources in such a way that the velocity component of the water that approaches perpendicular to the wall corresponds to the amount of water that is displaced due to the vibration movement.
In case of potential flow without wave radiation, the water flows across the entire area, in-phase and proportional to the velocity of the vibrating structure. The relation between vibration velocity and potential at the surface of the vibrating structure appears to be a measure of the added water mass. This relation is the same, both in case of a permanent flow (in which the water displacement due to the vibrating object has been replaced by sources that generate a permanent discharge) and in case of an oscillating flow.
Potential flow is based on very simplified equations for the water, i.e. the requirement that there is irrotational flow and the continuity condition.
If u, v and w are the velocity components in the directions x, y and z, and if p is the pressure, then it follows that:
1 1 1 , p u p v p w and x ρ t y ρ t z ρ t ∂ ∂ ∂ ∂ ∂ ∂ = − = − = − ∂ ∂ ∂ ∂ ∂ ∂ (C3.2)
The (velocity) potential Φ(x,y,z,t) is defined as:
u
x
∂Φ = −
∂ (C3.3)
2 2 2 2 2 2 0 u v w x y z x y z ∂ ∂ ∂ ∂ Φ ∂ Φ ∂ Φ + + = = − − − ∂ ∂ ∂ ∂ ∂ ∂ (C3.4)
From Equations C3.2 and C3.3 it follows that:
p constant t ρ∂Φ = + ∂ (C3.5) because: 2 p u x ρ x t t ∂ = ∂ Φ = −∂ ∂ ∂ ∂ ∂
From a potential flow calculation, the potential along the wall of the vibrating structure may be found, according to:
gives structure S
v Φ (3.6a)
in which vstructure = the velocity of the vibrating structure, and ΦS = the potential found at the wall of the structure. It also holds that:
results in structure s v t t ∂ ∂Φ ∂ ∂ (3.6b) structure v t ∂
∂ is the acceleration of the structure, and for the potential flow in Equation C3.5 it was found that
t
∂Φ
∂ results in the pressure. For that reason the ratio is:
s structure
v
α = Φ (3.7)
apart from ρfluid, it is also a measure for the relation between the pressure generated per unit of acceleration. After totalizing over the contours of the surface (and taking into account that only the force in the direction of the vibration is relevant), this therefore results in a measure for the volume of the added water mass. α has the dimension of a length (m).
3.2.3 Assessment of added water mass based on
schematized flow pattern
In Part A, Paragraph 3.2.6, a number of cases were discussed, in which a nearly closed gate was included as part of a wall. If the gate vibrates at right angles to the wall, a first
assessment of the added water mass may be carried out by assuming that the water approaches and runs off radially. This is illustrated in Part A, see Figures A3.9 and A3.10. The sector angle at which the water approaches and runs off, is determined by the geometry; in case of a gate as part of an infinite wall, the sector angle even becomes 180º. For two-dimensional real situations, the area in which the flow is coupled to the vibration of the gate is always limited. If not, an infinitely great added water mass would be found. The location of the free water surface relative to the structure provides a measure for the limitation of the flow area.
In case of flow-surrounded objects there is no radial approach flow or runoff and a method of assessment as indicated above therefore is not useful in those cases.
For two-dimensional situations such as often encountered at gates and valves, a potential calculation may be carried out according to the relaxation method. First, a potential of Φ = 0 is established (moreover, this might as well be any random value, even with an irregular distribution across the area). If the right conditions have been introduced along the edges, it is now possible to determine the eventual potential distribution by systematically determining, point by point, the local potential based on the potentials of the surrounding points that are temporarily kept at fixed values. By repeating this process again and again, the ideal potential is approximated more closely; this process is convergent. This calculation method is also suitable for flow-surrounded objects in wide water surfaces.
The calculation method in its complete form may be found in Kolkman (1988), and will be presented in the following paragraph for cases with an infinitely long strip in wide water that oscillates at right angles to its plane.
This example has been selected as it is simple, and the outcome may be verified directly. In the description of the calculation program that has been included in Appendix III, this method has been made more concrete.
This calculation procedure is also very well suited when using a spreadsheet.
3.2.4 Examples of complete calculations
Below, two examples are presented, the results of which may be verified with the data from an analytical calculation: the flat strip in wide water in case of translatory and rotational movements (from Kolkman (1988)). This calculation method is discussed in great detail, as it lends itself to being widely used.
a. Flat strip in wide water in case of translatory vibration
Figure C3.1 shows the points of calculation for the calculation of the strip.
Figure C3.1:
The total grid with conditions at the outer contours and at the symmetry and anti-symmetry lines.
For reasons of symmetry and anti-symmetry it may suffice to calculate within one quadrant. The symmetry lines may be replaced by a fixed wall, and the anti-symmetry lines by a contour along which the pressure remains constant. The last precondition is identical to that in case of a free water surface.
All contours, also those of the structure, are positioned in the middle between the points of calculation. The strip height is 2a. The vibration is represented by x x= ˆ sin( )ωt .
For the grid, a distance measure ΔL is chosen, that is equal in directions x and y. Figure C3.2 shows the handling of a sloping surface that vibrates.
Figure C3.2:
Discharges along the contours of a vibrating structure in case of a sloping wall.
For the vibration velocity, a random value V0 is introduced. As each calculational node is representative of an area with height and width ΔL, the edge discharge of the vibrating object equals:
0 q V L
Δ = Δ (C3.8) The calculation area consists of the first quadrant of Figure C3.1, see Figure C3.3.
Figure C3.3:
The calculation area with definitions used.
The dimensions of the area are H in horizontal direction and V in vertical direction. The calculation program uses the numbers NH and NV. The horizontal dimension therefore is H = NHΔL, and the vertical dimension is V = NVΔL. The strip height is NSΔL. In the
calculation, the potential is represented by the magnitude Φ. As already mentioned before, with this, Φ multiplied with the density of the fluid, ρ, after division by the vibration velocity, results in a measure for a part of the added water mass.
For the central area, the continuity condition that is used for the adjustment of each of these points applies, saying that the small discharges that come in from below, from above, from the left and from the right, together must equal zero.
For each of the small horizontal discharges, in case of an infinitely thin strip, it holds that: X q V L L L ΔΦ Δ = Δ = Δ = ΔΦ Δ (C3.9)
For the potential on the vibrating wall, relative to the potential of the calculational node nearby, it holds that:
1 1 2 2 q L q L Δ ⎛ ⎞ ΔΦ = ⎜ Δ ⎟= Δ Δ ⎝ ⎠ (C3.10)
In reality this always appears to result in a potential that is too great, which is the reason why, after many attempts, it has been decided to introduce a strip with thickness 0.3 ΔL (relative to the central axis). Therefore:
0
0.2q
ΔΦ = (C3.11)
In case of the relaxation procedure described above, it appears to be possible to reach a faster convergence by using the so-called ‘over-relaxation’. This means that the change of potential before and after the adjustment, will still be subjected to a multiplication coefficient. The over-relaxation is defined as being this coefficient. Empirically, within the framework of this calculation procedure, it has been determined that the over-relaxation results in a
substantial advantage as regards the number of times that the adjustment needs to be repeated. The over-relaxation should not be greater than approximately 1.7, otherwise the calculation process becomes instable.
The calculation process is stopped by using a criterion for stopping, described with the concept of ‘accuracy’. By starting off with the situation in which all potentials are zero, and therefore also the added water mass mw is zero, after all points of calculation have been released once, a steadily improving assessment of the added water mass is obtained. By repeating this procedure n times, the average improvement that is obtained may be described as mw/n. The improvement that is obtained with the last repetition, is compared with the average improvement. This ratio is referred to as ‘accuracy’. See also Figure C3.4.
Figure C3.4:
The applied criterion for stopping.
The adjustment of the points does not occur in the same way for all points. Therefore Figure C3.3 presents different categories.
CATEGORY 1: these are the central points, for which the continuity condition results in:
(Φabove− Φ + Φ) ( right − Φ + Φ) ( under − Φ + Φ − Φ = (3.12) ) ( left ) 0 which results in:
( a r u l) / 4
Φ = Φ + Φ + Φ + Φ (3.13)
in which Φ = potential in the corresponding calculational node.
CATEGORY 2: the angular element top left. The left and upper limits both have as a
precondition Φ = 0. The four discharges in the calculational node together again result in zero:
2(0− Φ + Φ − Φ + Φ − Φ +) ( r ) ( u ) 2(0− Φ = (3.14) ) 0
The factor ‘2’ results from the distance of the calculational node to the contour that is only ½ΔL. Therefore, the corresponding potential difference results in a discharge that is two times greater. After elaboration, C3.14 results in:
( r u) / 6
Φ = Φ + Φ (3.15)
CATEGORY 3: the elements on the left, situated above the strip. For these, it is found in an analogous way, that:
( a r u) / 5
Φ = Φ + Φ + Φ (3.16)
CATEGORY 4: the elements at the left side of the vibrating strip (except for the angular point bottom left). At height z, the resulting discharge through the wall equals q(z).
Based on the continuity condition, it is found that:
(Φ − Φ + Φ − Φ + Φ − Φ +a ) ( r ) ( u ) q z( ) 0= (3.17) which results in:
{
( a r u q z( ) / 3}
Φ = Φ + Φ + Φ + (3.18)
Without further deduction, for the remaining categories only the results are presented. CATEGORY 5: the element bottom left.
( a r q(1)) / 2
Φ = Φ + Φ + (3.19)
In this, q(l) is the discharge produced by the vibrating strip in the lower element. CATEGORY 6: the elements at the bottom, except for the angular points.
( a r l) / 3
Φ = Φ + Φ + Φ (3.20)
CATEGORY 7: the point bottom right.
( a l) / 4
CATEGORY 8: the point at the right, except for the angular points. ( a u l) / 5
Φ = Φ + Φ + Φ (3.22)
CATEGORY 9: the element top right.
( u l) / 6
Φ = Φ + Φ (3.23)
CATEGORY 10: the elements at the surface, except for the angular points. ( r u l) / 5
Φ = Φ + Φ + Φ (3.24)
Appendix III describes a calculation program, including the resulting outcome.
In Part A it was found that for a long flat strip the added water mass equals the density of the fluid, multiplied with the volume of the cylinder concerned (Figure A3.3). In Appendix III it was found that the water mass in one quadrant is 0.799 R2 (with R = radius of the circle concerned), while in theory for the quarter circle this would have been 0.785 R2. The grid that was investigated contained 54 by 18 points, of which the (bisected) strip was 6 points high. When more points are used, the accuracy increases.
b. Flat strip in wide water in case of rotational vibration
Also, a program has been developed that calculates the added polar water mass. For the greater part, this program equals that of the translatory strip, see Appendix III. The changes relate to the following points.
• The discharges per element, qs(Nz) are not equal along the entire height of the strip, but increase with the distance to the bottom (for this, {ω*(Nz-½)*ΔL}*ΔL is introduced (in the latter relation, Nz is a counting unit, in which NsΔL results in the strip height). The factor ½ relates to the position of the centre of the element concerned; ω is the angular velocity of the vibration.
• To determine the moment, the pressure that is found is multiplied again by (Nz -½)*ΔL*ΔL.
The moment that is eventually found (divided by ω), is made dimensionless with the strip height to the power of four.
In Appendix III it is indicated that in case of a rotating strip, the theoretically expected added (polar) water mass in one quadrant should be π/32 (= 0.0982), while the calculation program indicates 0.1046, this again in case of a grid of 54 by 18 calculational nodes and a (bisected) strip height of 6 points.
3.3 Calculations in the frequency domain
Calculations in the frequency domain are carried out in case of harmonic loads. If the harmonic load exerts a long-lasting influence, eventually a vibration is generated with the same frequency as that of the load, and with a constant amplitude. Precondition however is that the system is linear. If the external load makes contact at a certain moment in time, then there are also starting forces that damp out after a while. The eventually resulting vibration is
referred to as the vibration relating to the particular (or special) solution of the vibration equation, while at the beginning of the load there is superposition of the particular solution and the general solution. The general solution concerns the free (damping out) vibration that occurs after an initial load, that is no longer present after a while.
This paragraph offers a brief summary of the calculation methods that are available to determine the response in case of periodic load of a linear system. The point of departure for the single mass spring system, is a system with (linear) damping; in case of the multiple system an undamped system suffices. See also Bouma (1976).
3.3.1 The single (degree of freedom) mass spring system
Figure C3.5:
Diagram of a single (degree of freedom) mass spring system.
In case of a periodic load, the system in Figure C3.5 may be described with the following equation: 2 0 2 d d d d i t y y m c ky F e t t ω + + = (C3.25)
For the symbols used, see Part A, Paragraphs 2.2.1 and 2.2.2.
In this, it is assumed that the harmonic load F e0 i tω exerts a long-lasting influence and that the starting forces have damped out. The solution of Equation C3.25 in that case provides the particular solution.
The natural frequency of this system results from the solution y Y e= 0 i tωn of the reduced
2 2 d 0 d y m ky t + = (C3.26)
For the natural (angular) frequency ωn it is now found that:
n k m
ω = (C3.27)
The solution of Equation C3.25 may be presented in various ways:
( ) 0 or: (cos sin ) i t i t i t y Ye Y e y Y i e ω ω α ω α α + = = = + (C3.28)
Y is a complex number, Y0 is a real number. The phase angle, α, may result from the real part R and the imaginary part I of the solution Y, i.e.:
( / )α =arctg I R (C3.29)
Filling in y Ye= i tω in Equation C3.25 for the particular solution, results in:
0 2 F Y mω icω k = − + + (C3.30)
Multiplying the numerator and the denominator by the added complex value of the denominator, and introducing the natural angular frequency ωn and the relative damping γ (γ = cωn/2k or γ = c/(2mωn)) results in Equation C3.31. The relative damping results from relating the damping to the ‘critical damping’. The latter is the damping exactly at the point in which the system that experiences decaying vibration has no periodic component anymore (compare Equation C3.37). 2 2 2 2 2 0 2 2 1 2 1 4 n n n n i kY F ω γ ω ω ω ω γ ω ω ω ⎛ ⎞ − − ⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠ (C3.31)
Elaboration of y results in:
2 2 0 2 2 2 2 2 1 2 1 4 n n n n i F y k ω γ ω ω ω ω γ ω ω ω ⎛ ⎞ − − ⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠ (C3.32)
The phase angle α may now be calculated directly using Equation C3.29. The amplitude of Y is obtained by totalizing the squares of the real and the imaginary part of Y and extracting the square root of this. This is not further elaborated here. The presentation of the response calculation in the frequency domain may be found in Part A, Figure A2.2.
If the excitation has not exerted its influence for very long, but only begins at the moment in time t = 0, then also the general solution needs to be added to the solution of Y that was found (the so-called particular solution), in order to comply with the conditions at t = 0, i.e. that the deflection in that case, as well as the vibration velocity, are still zero.
The general solution (i.e. of the so-called natural process) results from the equation for the free vibration:
2 2 d d 0 d d y y m c ky t + t + = (C3.33)
The solution of this equation is a damped vibration that may be presented as: , ( )
rt pt i t
y Ye= =Y eω r= +p iω (C3.34)
in which ω still needs to be determined.
The real part of r indicates the extent of damping out of the vibration, the imaginary part of r indicates the periodicity.
Filling in Equation C3.34 in Equation C3.33 results in the following for r:
2 0
mr + + = (C3.35) cr k
which results in:
2 2 2 4 c c k r m m m = − ± − (C3.36)
The term under the radical sign results in zero exactly when (by definition) the damping is critical. Therefore the damping that was made dimensionless (the so-called relative damping), γ, is also defined in such a way that it equals 1 exactly when the damping is critical. or (because equation C3.27) 2 2 n c c m km γ γ ω = = (C3.37)
and that is exactly the damping at which the response is periodic or not.
If the term under the radical sign is negative (as the damping generally is small, this is usually the case), after filling in the value of the dimensionless damping γ and the natural frequency ωn, through Equations C3.27 and C3.37, this results in:
2
1 n n
r= −γω ±iω −γ (C3.38)
