Frequencydomain and stochastic model for an articulated wave power device


 Alyson Marsh
 1 years ago
 Views:
Transcription
1 Frequencydomain stochastic model for an articulated wave power device J. Cândido P.A.P. Justino Department of Renewable Energies, Instituto Nacional de Engenharia, Tecnologia e Inovação Estrada do Paço do umiar, 9 isboa, Portugal Abstract To have the first look into device performance, analytical numerical tools must be used. Assuming that the wave power system hydrodynamics has a linear behaviour, diffraction radiation coefficients can be computed. If the power takeoff equipment may be, for the first approach, regarded as holding a linear behaviour then overall (i.e. hydrodynamic plus mechanical device performance can be studied for regular waves. In this study a frequencydomain model describes the articulated system behaviour for regular waves. For this paper a stochastic model is found for an articulated wave power device, probability density functions are defined for the relevant parameters that characterize the wave power system behaviour. For these parameters for different sea states the probability density functions are found. The articulated system is characterized by these probability density functions. Also, average values for capture width are obtained for these sea state conditions. Keywords: Wave energy, frequencydomain, stochastic model. Nomenclature A ij Â = added mass hydrodynamic coefficient = complex wave elevation amplitude Â n = complex rom amplitude B ij = damping hydrodynamic coefficient C i = hydrostatic restoring coefficient for body i D = damping coefficient of the power takeoff equipment E {} = expected value of F D i = complex amplitude for the diffraction force on body i F = load force HG, HG = transfer function H s = significant wave height K = spring coefficient of the power takeoff equipment M i = mass of body i P I = incident power for a regular wave P u = average useful power S η = spectral density T = time interval T e = wave energy period Z = load impedance g = acceleration of gravity h = water depth k = wave number t = time ϕ n = rom phase η = sea surface elevation λ c = capture width θ = wave direction angle = water specific mass ρ σ = variance ξ i = complex amplitude displacement for body i * = complex conjugate Proceedings of the 7th European Wave Tidal Energy Conference, Porto, Portugal, 7 Introduction Since the beginning of the th century man has been thinking on how to make use of wave power. So far more than about thouss of patents have been registered. To have the first look into device performance, analytical numerical tools must be used. Assuming that the wave power system hydrodynamics has a linear behaviour, diffraction radiation coefficients can be computed using, for example, WAMIT or Aquadyn. If the power takeoff equipment can be, in the first approach, regarded as holding a linear behaviour, then overall (i.e. hydrodynamic plus mechanical device performance can be studied for regular waves. In this study a frequencydomain
2 model describes the behaviour for regular waves of an articulated system consisting of two main concentric bodies. Optimal mechanical damping spring coefficients are computed for some parameters (relative absolute displacements restrictions. Useful power, capture width, other variables, like relative displacements (displacement of one body for a coordinate reference system that oscillates with the same amplitude phase as the other body, i.e. a coordinate reference system fixed with respect to the second body, are computed for regular waves the mechanical coefficients mentioned above. In [] a theory for wave power absorption by two independently oscillating bodies has already been devised. Frequency analysis has been used to study devices such as Searev wave energy converter, [], also the hydrodynamic performance of arrays of devices as in []. As mentioned before software using panel methods can be used to compute the hydrodynamic coefficients of the two concentric bodies. In [] hydrodynamic coefficients in heave of two concentric surfacepiercing truncated circular cylinders were already computed. A stochastic model has already been developed for OWC power plants []. This stochastic model has been used for optimization procedures of the FOZ do Douro OWC plant, [], [7]. In this paper a stochastic model is found for the articulated wave power device, probability density functions are defined for the relevant parameters that characterize the wave power system behaviour. Assuming that the overall system behaviour is linear that the wave elevation for the irregular waves may be regarded as a stochastic process with a Gaussian probability density function, the variables that define the system behaviour, like, for example, displacements for the articulated system elements, will also have a Gaussian probability density function. For these parameters for different sea states the probability density functions (i.e. variances are found. The articulated system is characterized by these functions. Also, average values for capture width are obtained for these sea state conditions. Mathematical models The wave energy device is made of two concentric axisymmetric oscillating bodies, Fig.. The relative heave motion between bodies allows extracting power from sea waves. As usual it is assumed that the bodies have linear hydrodynamic behaviour also that the power takeoff can be modelled by spring damping terms proportional to the relative displacement between bodies to the relative velocity, respectively. A frequency domain model a stochastic model are subsequently presented. Frequency domain model Applying Newton s second law, the governing equations for the wave energy device are, M ( ( ˆ ω ξ = ω A iωb ξ + ω A iωb ξ + FD Cξ ( K + iωd ( ξ ξ, (, M ( ( ˆ ω ξ = ω A iωb ξ + ω A iωb ξ + FD Cξ + ( K + iωd ( ξ ξ, ( where ω is the angular frequency, ξ i is the complex amplitude displacement for body i, M i the mass of body i, C i the hydrostatic restoring coefficient for body i, Aij B ij the added mass damping hydrodynamic coefficients, F D the complex amplitude for the diffraction i force on body i, K D the spring damping coefficients of the power takeoff equipment. As given in [] for the angular frequency ω the average useful power can be given by, P ˆ ˆ u = Dω ξ ξ. ( Thus, choosing values for K D it is possible to compute for each angular frequency wave elevation amplitude the heave motions of the two concentric floating bodies, ξ ˆ ξ, as well as the average useful power, P u. The capture width, λ c, may be computed by, ˆ ˆ P Dω ξ ξ u λ c = =, ( PI ˆ kh ρg A( ω + tanh( kh sinh( kh where P I is the incident power for a regular wave with angular frequency ω complex wave elevation amplitude Â ( ω. The water depth is h, k is the wave number given by the positive root of the dispersion relationship, ω = k tanh( kh, ρ is the water specific g mass g the acceleration of gravity. Stochastic model As in [] we will consider a time interval of duration T, assume that the sea surface elevation, η ( t, is a Gaussian rom variable given by, ( ˆ η t = An exp( inωt, ( n= where ω = π / T, Aˆ n = Aˆ n exp( iϕ n is a complex rom variable with ϕ n being a rom variable uniformly distributed in the interval [ ;π [ assuming that Aˆ E n = σ n { ˆ * ' } = E A n A n, for n n'.
3 Following [] we find for the variance of the sea surface elevation, assuming that the sea state can be represented by a discrete power spectrum, * { } exp( ( ' { ˆ ˆ * σ η = E ηη = i n n ωt E An A ' } = n n= n' = = σ n. n= ( As it is well known, if the sea surface power spectrum is continuous the variance of the elevation is given by ση = S η ( ω dω, (7 S η ω is a spectral density defined in the range where ( ] [ ;. Thus in the limiting case when T we will get ω dω E Aˆ n = σ n Sη ( ω dω. If the power spectrum is dependent not only on the wave frequency but also on the direction of the incoming waves we get for ( M ( ˆ η t = Anm exp( inωt, ( m= n= where M is the number of considered directions for the interval [ ;π [, Aˆ E nm = σ nm { ˆ * ' ' } = E A nm A n m for n n' m m'. The variance of the elevation for a continuous power spectrum is now π ση = S η ( ω, θ dωdθ. (9 Taking into consideration that the two oscillating bodies are axisymmetric that their behaviour in the frequency domain can be described by eqs. ( (, it is possible to find transfer functions, HG ( nω HG ( nω, that relate the amplitude of the incident wave A n the displacement amplitude for body, ξ ( nω = HG ( nω An, (, ξ ( nω = HG ( nω Aˆn. ( According to ( the vertical displacements for bodies are described by, ( t = HG ( nω Aˆ exp( inω t, ξ n= n ( ( t = HG ( nω Aˆ exp( inω t. ξ n ( n= Note that just as η, also ξ ξ are Gaussian rom variables, with variances * { ξ ξ } = HG ( nω σ, σ ξ = E n ( n= * { } σ ξ = E ξ ξ ( = HG nω σ n. ( n= If the sea state can be represented by a continuous power spectrum, then we get for ξ ξ variances σ = ( (, S η ω HG ω dω ( ξ σ = ( (. S η ω HG ω dω (7 ξ It is straightforward that the variance for the velocity of bodies may be computed by, σ & = Sη ( ω ω HG ( ω dω, ( ξ σ & = Sη ( ω ω HG ( ω dω. (9 ξ Assuming that the load force, F, can be given by ( exp( inω t, ( ˆ F t = ± ( K + inωd ξ( nω ξ ( nω n= ( we get, for the variance of the load force, * σ = E{ FF } = F * * ( ( ' ( ˆ ˆ ( ˆ ˆ = + Z nω Z n ω E ξ ξ n ξ ξ n' n= n' = exp( i( n n' ωt, ( with Z ( nω = K + inωd. Since ξ ξ are given by eqs. ( E{ ( ξ ξ n ( ξ ξ n' } = for n n', we get for the variance of F σ = Z ( nω HG ( nω HG ( nω σ n. ( F n= Thus, for a continuous power spectrum we may write σ = Sη ( ω Z ( ω HG ( ω HG ( ω dω. ( F The average useful power may be written as P = DE & u ξ ( t & ξ ( t, (
4 taking into consideration that for a sea state represented by a continuous power spectrum we get E & ξ & ξ = E (& ξ & ξ (& ξ & ξ * = ( = Sη ( ω ω HG ( ω HG ( ω dω. Note that the probability of the instantaneous useful power being less or equal than χ D is given by Prob χ x χ exp dx. χ E & ξ & ξ ( P u D = Numerical results ( To illustrate the application of the frequencydomain model mainly the stochastic model, results were obtained for the oscillating body presented in Fig.. It is assumed that body is the body with a toroidal shape (outside body body is the inside body made of two parts (one surfacepiercing body a completely submerged cylinder that oscillate together. WAMIT was used to compute the hydrodynamic diffraction radiation coefficients for a set of wave frequencies in the range of.9 rad/s to.77 rad/s. The water depth is m. Assuming two scenarios for the power takeoff equipment: a it can only be simulated by a damping, D, coefficient, b it can be simulated by spring, K, damping coefficients, values were computed for K D that maximize the average useful power, P u, thus the capture width, λ c. Several restrictions were also imposed on the amplitude of the heave motion of body on the amplitude of the relative heave motion between body body, ξ ξ, for an incident wave with amplitude of m. For scenario a the capture width, mechanical damping coefficient, relative amplitude displacement between bodies absolute amplitude displacements are presented in Figs. to, respectively. They are obtained for different displacement restrictions. In all the cases it is assumed that the amplitude for the relative displacement between body body cannot exceed m for an incident wave of m amplitude. It is also considered that the absolute displacement amplitude for body cannot exceed,,, 7 m. Thus the case _ means that the amplitude for the relative displacement cannot exceed m the amplitude for the absolute displacement cannot be greater than m. It can be observed from Fig. that the device has two well defined capture width peaks, one at.s another at.7s. The value for the average useful power for the second peak (at.7s is closely related to the allowed maximum amplitude heave motion for the first body, Fig.. To be able to control this amplitude, appropriate mechanical damping coefficients need to be chosen, Fig.. Note that if there are restrictions on the relative displacement between bodies on the absolute displacement of body then there will be restrictions for the absolute displacement of body, implicitly. For scenario b, Figs. 7 to present capture width, mechanical damping spring coefficients, relative amplitude displacement absolute amplitude displacements, respectively. Again, in all the cases it is assumed that the amplitude for the relative displacement between body body cannot exceed m that the absolute displacement amplitude for body cannot exceed, m. As expected, for different maximum amplitude displacements of body we get different capture widths for lower values for body maximum absolute displacement amplitude we get smaller values for capture width Fig. 7. Indeed, damping spring coefficients must be tuned out to allow for the restrictions to be achieved, Figs. 9. It may be observed from Fig. that the absolute displacement amplitude for body is bounded, as well as the relative displacement amplitude between bodies, Fig.. For the stochastic model results are shown for irregular waves. Again, the two scenarios for the power takeoff equipment above explained were considered. The spring damping coefficients that maximize eq. (, thus the respective capture width, λ c, were computed. Results are obtained for the variance of ξ ξ, as well as for the variance of the load force, F. The best K D for each sea state are also shown. The variances for ξ, ξ F were obtained for sea states with H s, the sea state significant wave height, equal to m. Knowing that σ, σ σ are given by eqs. (, (7 (, ξ ξ F respectively, it is easily found that the variances are proportional to the square of H s, i.e., if we multiply H s by, the variances will be multiplied by, assuming that T e, the sea state wave energy period, is kept constant as well as K D. To represent the sea states the following frequency spectrum was adopted ([9]:   S η ( ω =.Hs Te ω exp( Te ω. (7 The device behaviour was simulated for wave energy periods from 7 to s for a H s equal to m. As already mentioned, two different scenarios were considered for the power takeoff equipment. Figs. to present the capture width, mechanical damping, spring coefficient variances for the displacements of body, load force, for both power takeoff scenarios, respectively. It can be observed that the system has a better performance for sea states with smaller wave energy period, Fig.. For the range of considered wave energy periods the damping coefficients for scenario a are greater than the ones computed for scenario b, Fig.. The spring coefficients computed for scenario b may be negative or positive,
5 depending on the wave energy period, Fig.. For scenario a the curves for the displacement variance of both bodies are similar, Fig.. However, these curves are rather different for scenario b, Fig. 7. The variance for the load force converges for both scenarios when the wave energy period increases, Fig.. As the capture width is quite different for T e equal to 7s for the considered scenarios, the variance for the load force should also be different as it is observed in Fig.. The obtained variances allow defining Gaussian probability density functions for the analyzed parameters. th European Wave Tidal Energy Conference, Glasgow,. [] J. Falnes. Ocean waves oscillating systems. Cambridge University Press. [9] Y. Goda. Rom Seas Design of Maritime Structures. University of Tokyo Press 9. Conclusions Frequencydomain stochastic models were devised for an articulated device composed by two concentric bodies. Results were obtained for regular irregular waves. The use of the stochastic model for irregular waves allows to find variances that define Gaussian probability density functions for relevant wave device parameters. It was assumed that the power takeoff mechanical equipment has a linear behaviour can be modelled by spring damping coefficients. For irregular waves it was assumed that the characteristics of the power takeoff system are constant for the duration of a sea state. As expected, the capture widths for the stochastic model are significantly lower than the ones obtained for the frequencydomain model. In order to have better performances for irregular waves, device control will be essential. References [] M.A. Srokosz D.V. Evans. A theory for wavepower absorption by two independently oscillating bodies. J. Fluid Mechanics, 9(:7, 979. [] A. Clément, A. Babarit, JC. Gilloteaux, C. Josset G. Duclos. The SEAREV wave energy converter. In Proc. th European Wave Tidal Energy Conference, Glasgow,. [] P.A.P. Justino A. Clément. Hydrodynamic performance for small arrays of submerged spheres. In Proc. th European Wave Energy Conference, Cork,. [] S. Mavrakos. Hydrodynamic coefficients in heave of two concentric surfacepiercing truncated circular cylinders. Applied Ocean Research, :97,. [] A.F. de O. Falcão R.J.A. Rodrigues. Stochastic modelling of OWC wave power plant performance. Applied Ocean Research, :97,. [].M.C. Gato, P.A.P. Justino A.F. de O. Falcão. Optimization of power takeoff equipment for an oscillatingwater column wave energy plant. th European Wave Tidal Energy Conference, Glasgow,. [7] M.T. Pontes, J. Cândido, J.C.C. Henriques P. Justino. Optimizing OWC sitting in the nearshore. In Proc. (D Mar 7  Figure : Panel grid describing the wet surface of the concentric axisymmetric oscillating bodies in numerical evaluation. Capture Width (m T (s X Z Y _7 _ Figure : Capture width for a m maximum displacement amplitude between body body (for an incident wave of m amplitude maximum absolute displacement amplitude for body varying between m m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D.
6 Coefficient (Ns/m.E+.E+.E+.E+.E+.E+ _7 _ T (s Figure : Mechanical damping coefficient for a m maximum displacement amplitude between body body (for an incident wave of m amplitude maximum absolute displacement amplitude for body varying between m m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D. Relative Displacement Amplitude (m _7 _ T (s Figure : Relative displacement amplitude between body body (for an incident wave of m amplitude for maximum absolute displacement amplitude for body varying between m m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D. Body Displacement Amplitude (m 9 7 _7 _ T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, an absolute displacement amplitude restriction varying between m m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D. Body Displacement Amplitude (m 9 7 _7 _ T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, an absolute displacement amplitude restriction for body varying between m m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D.
7 Capture Width (m T (s Figure 7: Capture width for a m maximum displacement amplitude between body body (for an incident wave of m amplitude different maximum absolute displacement amplitudes for body of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K. Coefficient (Ns/m.E+.E+.E+.E+.E+ T (s Figure : Mechanical damping coefficient for a m maximum displacement amplitude between body body (for an incident wave of m amplitude different maximum absolute displacement amplitudes for body of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K. Spring Coefficient (N/m.E+.E+.E+ .E+ .E+ T (s Figure 9: Mechanical spring coefficient for a m maximum displacement amplitude between body body (for an incident wave of m amplitude different maximum absolute displacement amplitudes for body of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K. Relative Displacement Amplitude (m T (s Figure : Relative displacement amplitude between body body (for an incident wave of m amplitude for different maximum absolute displacement amplitudes for body of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K.
8 Body Displacement Amplitude (m 9 T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, absolute displacement amplitude restrictions of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K. Body Displacement Amplitude (m T (s Figure : Absolute displacement amplitude for body (for an incident wave of m amplitude, considering a m maximum displacement amplitude between body body, absolute displacement amplitude restrictions for body of m, m m, assuming that the power takeoff equipment can be simulated by damping spring coefficients D K. Capture Width (m.... Spring 7 9 Te(s Figure : Capture width for wave energy periods from 7 to s H s equal to m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D (black line assuming that the power takeoff equipment can be simulated by both damping spring coefficients D K (red line. Coefficient (Ns/m.E+.E+.E+.E+.E+ Spring 7 9 Te(s Figure : Mechanical damping coefficient for wave energy periods from 7 to s H s equal to m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D (black line assuming that the power takeoff equipment can be simulated by both damping spring coefficients D K (red line.
9 Spring Coefficient (N/m.E+.E+.E E+ .E+ Spring Te(s Figure : Mechanical spring coefficient for wave energy periods from 7 to s H s equal to m, assuming that the power takeoff equipment can be simulated by both damping spring coefficients D K. Displacement Variance (m Body Body 7 9 Te (s Figure 7: Variances for the displacements of body, for wave energy periods from 7 to s H s equal to m, assuming that the power takeoff equipment can be simulated by both damping spring coefficients D K. Displacement Variance (m Body Body 7 9 Te (s Figure : Variances for the displacements of body, for wave energy periods from 7 to s H s equal to m, assuming that the power takeoff equipment can only be simulated by a damping coefficient. oad Force Variance (N^.E+.E+.E+.E+.E+.E+.E+ Spring 7 9 Te(s Figure : Variance for load force, for wave energy periods from 7 to s H s equal to m, assuming that the power takeoff equipment can only be simulated by a damping coefficient D (black line assuming that the power takeoff equipment can be simulated by both damping spring coefficients D K (red line.
WavePiston Project. Structural Concept Development Hydrodynamic Aspects. Preliminary Analytical Estimates. February 2010
WavePiston Project Structural Concept Development Hydrodynamic Aspects Preliminary Analytical Estimates February 2010 WavePiston Project Structural Concept Development Hydrodynamic Aspects Preliminary
More informationWave Energy Absorption in Irregular Waves by a Floating Body Equipped with Interior Rotating ElectricPower Generator
Journal of Ocean and Wind Energy (ISSN 31364) Copyright by The International Society of Offshore and Polar Engineers Vol. 1, No. 3, August 14, pp. 19 134 http://www.isope.org/publications Wave Energy
More informationSimulation of Offshore Structures in Virtual Ocean Basin (VOB)
Simulation of Offshore Structures in Virtual Ocean Basin (VOB) Dr. Wei Bai 29/06/2015 Department of Civil & Environmental Engineering National University of Singapore Outline Methodology Generation of
More informationCFD SUPPORTED EXAMINATION OF BUOY DESIGN FOR WAVE ENERGY CONVERSION
CFD SUPPORTED EXAMINATION OF BUOY DESIGN FOR WAVE ENERGY CONVERSION Nadir Yilmaz, Geoffrey E. Trapp, Scott M. Gagan, Timothy R. Emmerich Department of Mechanical Engineering, New Mexico Institute of Mining
More informationSIMULATION OF A SLACKMOORED HEAVINGBUOY WAVEENERGY CONVERTER WITH PHASE CONTROL
SIMULATION OF A SLACKMOORED HEAVINGBUOY WAVEENERGY CONVERTER WITH PHASE CONTROL by Håvard Eidsmoen Division of Physics Norwegian University of Science and Technology N7034 Trondheim Norway May 1996
More informationSIMULATION OF A TIGHTMOORED AMPLITUDE LIMITED HEAVINGBUOY WAVEENERGY CONVERTER WITH PHASE CONTROL
SIMULATION OF A TIGHTMOORED AMPLITUDE LIMITED HEAVINGBUOY WAVEENERGY CONVERTER WITH PHASE CONTROL by Håvard Eidsmoen Division of Physics, Norwegian University of Science and Technology, N704 Trondheim,
More informationC06 CALM Buoy. Orcina. 1. CALM Buoy. Introduction. 1.1. Building the model
C06 CALM Buoy Introduction In these examples, a CALM buoy is moored by six equally spaced mooring lines. A shuttle tanker is moored to the buoy by a hawser, with fluid transfer through a floating hose.
More informationWave Hydro Dynamics Prof. V. Sundar Department of Ocean Engineering Indian Institute of Technology, Madras
Wave Hydro Dynamics Prof. V. Sundar Department of Ocean Engineering Indian Institute of Technology, Madras Module No. # 02 Wave Motion and Linear Wave Theory Lecture No. # 03 Wave Motion II We will today
More informationTHE PERFORMANCE OF A WAVE ENERGY CONVERTER IN SHALLOW WATER
6 th European Wave and Tidal Energy Conference Glasgow, UK August 9 th  September nd 5 THE PERFORMANCE OF A WAVE ENERGY CONVERTER IN SHALLOW WATER Matt Folley, Trevor Whittaker, Alan Henry School of Civil
More informationStatistical Analysis of Nonstationary Waves off the Savannah Coast, Georgia, USA
1 Statistical Analysis of Nonstationary Waves off the Savannah Coast, Xiufeng Yang School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA, 30332 (xfyang@gatech.edu)
More informationMODULE VII LARGE BODY WAVE DIFFRACTION
MODULE VII LARGE BODY WAVE DIFFRACTION 1.0 INTRODUCTION In the wavestructure interaction problems, it is classical to divide into two major classification: slender body interaction and large body interaction.
More informationJournal of Engineering Science and Technology Review 2 (1) (2009) 7681. Lecture Note
Journal of Engineering Science and Technology Review 2 (1) (2009) 7681 Lecture Note JOURNAL OF Engineering Science and Technology Review www.jestr.org Time of flight and range of the motion of a projectile
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationMECH 450F/580 COURSE OUTLINE INTRODUCTION TO OCEAN ENGINEERING
Department of Mechanical Engineering MECH 450F/580 COURSE OUTLINE INTRODUCTION TO OCEAN ENGINEERING Spring 2014 Course Web Site See the MECH 450F Moodle site on the UVic Moodle system. Course Numbers:
More informationSimplified formulas of heave added mass coefficients at high frequency for various twodimensional bodies in a finite water depth
csnak, 2015 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 http://dx.doi.org/10.1515/ijnaoe20150009 pissn: 20926782, eissn: 20926790 Simplified formulas of heave added mass coefficients at high frequency
More informationApplications of SecondOrder Differential Equations
Applications of SecondOrder Differential Equations Secondorder linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationSolution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k
Physics 1C Midterm 1 Summer Session II, 2011 Solutions 1. If F = kx, then k m is (a) A (b) ω (c) ω 2 (d) Aω (e) A 2 ω Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationUpdated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum
Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More information2. The graph shows how the displacement varies with time for an object undergoing simple harmonic motion.
Practice Test: 29 marks (37 minutes) Additional Problem: 31 marks (45 minutes) 1. A transverse wave travels from left to right. The diagram on the right shows how, at a particular instant of time, the
More informationMODEL BASED DESIGN OF EFFICIENT POWER TAKEOFF SYSTEMS FOR WAVE ENERGY CONVERTERS
The Twelfth Scandinavian International Conference on Fluid Power, May 182, 211, Tampere, Finland MODEL BASED DESIGN OF EFFICIENT POWER TAKEOFF SYSTEMS FOR WAVE ENERGY CONVERTERS Rico H. Hansen, Torben
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationOpen channel flow Basic principle
Open channel flow Basic principle INTRODUCTION Flow in rivers, irrigation canals, drainage ditches and aqueducts are some examples for open channel flow. These flows occur with a free surface and the pressure
More informationABS TECHNICAL PAPERS 2008 A STERN SLAMMING ANALYSIS USING THREEDIMENSIONAL CFD SIMULATION. Suqin Wang Email: SWang@eagle.org
ABS TECHNICAL PAPERS 8 Proceedings of OMAE 8 7 th International Conference on Offshore Mechanics and Arctic Engineering 15 June, 8, Estoril, Portugal OMAE85785 A STERN SLAMMING ANALYSIS USING THREEDIMENSIONAL
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationScienceDirect. Pressure Based Eulerian Approach for Investigation of Sloshing in Rectangular Water Tank
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 144 (2016 ) 1187 1194 12th International Conference on Vibration Problems, ICOVP 2015 Pressure Based Eulerian Approach for Investigation
More informationRANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA
RANDOM VIBRATION AN OVERVIEW by Barry Controls, Hopkinton, MA ABSTRACT Random vibration is becoming increasingly recognized as the most realistic method of simulating the dynamic environment of military
More informationTime domain modeling
Time domain modeling Equationof motion of a WEC Frequency domain: Ok if all effects/forces are linear M+ A ω X && % ω = F% ω K + K X% ω B ω + B X% & ω ( ) H PTO PTO + others Time domain: Must be linear
More informationFig. 9: an immersed body in a fluid, experiences a force equal to the weight of the fluid it displaces.
Buoyancy Archimedes s 1 st laws of buoyancy: A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces, see Fig. 9 and 10. Fig. 9: an immersed body in
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationCE 204 FLUID MECHANICS
CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 TuzlaIstanbul/TURKEY Phone: +902166771630 ext.1974 Fax: +902166771486 Email:
More informationSection 5.0 : Horn Physics. By Martin J. King, 6/29/08 Copyright 2008 by Martin J. King. All Rights Reserved.
Section 5. : Horn Physics Section 5. : Horn Physics By Martin J. King, 6/29/8 Copyright 28 by Martin J. King. All Rights Reserved. Before discussing the design of a horn loaded loudspeaker system, it is
More informationSOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS
SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS  VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering
More information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonlyoccurring firstorder and secondorder ordinary differential equations.
More informationSimple Harmonic Motion
Simple Harmonic Motion Restating Hooke s law The equation of motion Phase, frequency, amplitude Simple Pendulum Damped and Forced oscillations Resonance Harmonic Motion A lot of motion in the real world
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
More informationPHYS 1014M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PHYS 1014M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in
More informationStudy for the Hull Shape of a Wave Energy ConverterPoint Absorber
Study for the Hull Shape of a Wave Energy ConverterPoint Absorber Design Optimization & Modeling Improvement Filippos Kalofotias A thesis presented for the degree of Master of Science Water Engineering
More informationA C O U S T I C S of W O O D Lecture 3
Jan Tippner, Dep. of Wood Science, FFWT MU Brno jan. tippner@mendelu. cz Content of lecture 3: 1. Damping 2. Internal friction in the wood Content of lecture 3: 1. Damping 2. Internal friction in the wood
More informationDetermination of Acceleration due to Gravity
Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two
More informationFirst Power Production figures from the Wave Star Roshage Wave Energy Converter
First Power Production figures from the Wave Star Roshage Wave Energy Converter L. Marquis 1, M. Kramer 1 and P. Frigaard 1 Wave Star A/S, Gammel Vartov Vej 0, 900 Hellerup, Denmark Email: info@wavestarenergy.com
More informationModelling and Control of the Wavestar Prototype
Modelling and Control of the Wavestar Prototype Rico H. Hansen Energy Technology, Aalborg University & Wave Star A/S Pontoppidanstraede 11, 9 Aalborg, Denmark Email: rhh@wavestarenergy.com Morten M. Kramer
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationSOFTWARE FOR GENERATION OF SPECTRUM COMPATIBLE TIME HISTORY
3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 6, 24 Paper No. 296 SOFTWARE FOR GENERATION OF SPECTRUM COMPATIBLE TIME HISTORY ASHOK KUMAR SUMMARY One of the important
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More information2.016 Hydrodynamics Prof. A.H. Techet Fall 2005
.016 Hydrodynamics Reading #7.016 Hydrodynamics Prof. A.H. Techet Fall 005 Free Surface Water Waves I. Problem setu 1. Free surface water wave roblem. In order to determine an exact equation for the roblem
More informationThe Fourier Analysis Tool in Microsoft Excel
The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier
More informationDetermination of source parameters from seismic spectra
Topic Determination of source parameters from seismic spectra Authors Michael Baumbach, and Peter Bormann (formerly GeoForschungsZentrum Potsdam, Telegrafenberg, D14473 Potsdam, Germany); Email: pb65@gmx.net
More informationExamples of Uniform EM Plane Waves
Examples of Uniform EM Plane Waves Outline Reminder of Wave Equation Reminder of Relation Between E & H Energy Transported by EM Waves (Poynting Vector) Examples of Energy Transport by EM Waves 1 Coupling
More informationStochastic Doppler shift and encountered wave period distributions in Gaussian waves
Ocean Engineering 26 (1999) 507 518 Stochastic Doppler shift and encountered wave period distributions in Gaussian waves G. Lindgren a,*, I. Rychlik a, M. Prevosto b a Department of Mathematical Statistics,
More informationENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION
ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION This tutorial covers prerequisite material and should be skipped if you are
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationSinking Bubble in Vibrating Tanks Christian Gentry, James Greenberg, Xi Ran Wang, Nick Kearns University of Arizona
Sinking Bubble in Vibrating Tanks Christian Gentry, James Greenberg, Xi Ran Wang, Nick Kearns University of Arizona It is experimentally observed that bubbles will sometimes sink to the bottom of their
More informationPhysics 207 Lecture 25. Lecture 25. For Thursday, read through all of Chapter 18. Angular Momentum Exercise
Lecture 5 Today Review: Exam covers Chapters 1417 17 plus angular momentum, statics Assignment For Thursday, read through all of Chapter 18 Physics 07: Lecture 5, Pg 1 Angular Momentum Exercise A mass
More informationResponse to Harmonic Excitation Part 2: Damped Systems
Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost
More informationAS COMPETITION PAPER 2008
AS COMPETITION PAPER 28 Name School Town & County Total Mark/5 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Marks allocated for each question
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationSeismic Analysis and Design of Steel Liquid Storage Tanks
Vol. 1, 005 CSA Academic Perspective 0 Seismic Analysis and Design of Steel Liquid Storage Tanks Lisa Yunxia Wang California State Polytechnic University Pomona ABSTRACT Practicing engineers face many
More informationSlide 10.1. Basic system Models
Slide 10.1 Basic system Models Objectives: Devise Models from basic building blocks of mechanical, electrical, fluid and thermal systems Recognize analogies between mechanical, electrical, fluid and thermal
More informationLecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2)
Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2) In this lecture How does turbulence affect the ensemblemean equations of fluid motion/transport? Force balance in a quasisteady turbulent boundary
More informationHomework #7 Solutions
MAT 0 Spring 201 Problems Homework #7 Solutions Section.: 4, 18, 22, 24, 4, 40 Section.4: 4, abc, 16, 18, 22. Omit the graphing part on problems 16 and 18...4. Find the general solution to the differential
More informationF en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself.
The Electron Oscillator/Lorentz Atom Consider a simple model of a classical atom, in which the electron is harmonically bound to the nucleus n x e F en = mω 0 2 x origin resonance frequency Note: We should
More informationVISCOSITY OF A LIQUID. To determine the viscosity of a lubricating oil. Time permitting, the temperature variation of viscosity can also be studied.
VISCOSITY OF A LIQUID August 19, 004 OBJECTIVE: EQUIPMENT: To determine the viscosity of a lubricating oil. Time permitting, the temperature variation of viscosity can also be studied. Viscosity apparatus
More informationManufacturing Equipment Modeling
QUESTION 1 For a linear axis actuated by an electric motor complete the following: a. Derive a differential equation for the linear axis velocity assuming viscous friction acts on the DC motor shaft, leadscrew,
More informationExperimental and numerical investigation of slamming of an Oscillating Wave Surge Converter in two dimensions
Experimental and numerical investigation of slamming of an Oscillating Wave Surge Converter in two dimensions T. Abadie, Y. Wei, V. Lebrun, F. Dias (UCD) Collaborating work with: A. Henry, J. Nicholson,
More informationFraunhofer Diffraction
Physics 334 Spring 1 Purpose Fraunhofer Diffraction The experiment will test the theory of Fraunhofer diffraction at a single slit by comparing a careful measurement of the angular dependence of intensity
More informationHow to compute Random acceleration, velocity, and displacement values from a breakpoint table.
How to compute Random acceleration, velocity, and displacement values from a breakpoint table. A random spectrum is defined as a set of frequency and amplitude breakpoints, like these: 0.050 Acceleration
More informationWave Motion (Chapter 15)
Wave Motion (Chapter 15) Waves are moving oscillations. They transport energy and momentum through space without transporting matter. In mechanical waves this happens via a disturbance in a medium. Transverse
More informationPHYSICAL QUANTITIES AND UNITS
1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them
More informationSTUDY OF DAMRESERVOIR DYNAMIC INTERACTION USING VIBRATION TESTS ON A PHYSICAL MODEL
STUDY OF DAMRESERVOIR DYNAMIC INTERACTION USING VIBRATION TESTS ON A PHYSICAL MODEL Paulo Mendes, Instituto Superior de Engenharia de Lisboa, Portugal Sérgio Oliveira, Laboratório Nacional de Engenharia
More information226 Chapter 15: OSCILLATIONS
Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion
More informationGravity  Geoscientific Measure: The variations in the Earth s gravitational field due to changes in density of materials within the Earth.
Gravity  Geoscientific Measure: The variations in the Earth s gravitational field due to changes in density of materials within the Earth. Objective Map variation in gravitational field associated with
More informationNumerical study on Sinking Mooring System of Immersed Tube Tunnel Construction
5th International Conference on Advanced Engineering Materials and Technology (AEMT 2015) Numerical study on Sinking Mooring System of Immersed Tube Tunnel Construction Qiu Zhengzhong 1, a * and Wan Meng
More informationStress and deformation of offshore piles under structural and wave loading
Stress and deformation of offshore piles under structural and wave loading J. A. Eicher, H. Guan, and D. S. Jeng # School of Engineering, Griffith University, Gold Coast Campus, PMB 50 Gold Coast Mail
More informationHOOKE S LAW AND SIMPLE HARMONIC MOTION
HOOKE S LAW AND SIMPLE HARMONIC MOTION Alexander Sapozhnikov, Brooklyn College CUNY, New York, alexs@brooklyn.cuny.edu Objectives Study Hooke s Law and measure the spring constant. Study Simple Harmonic
More informationSimple Pendulum 10/10
Physical Science 101 Simple Pendulum 10/10 Name Partner s Name Purpose In this lab you will study the motion of a simple pendulum. A simple pendulum is a pendulum that has a small amplitude of swing, i.e.,
More informationXI / PHYSICS FLUIDS IN MOTION 11/PA
Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A
More informationPhysical Quantities, Symbols and Units
Table 1 below indicates the physical quantities required for numerical calculations that are included in the Access 3 Physics units and the Intermediate 1 Physics units and course together with the SI
More informationSecond Order Systems
Second Order Systems Second Order Equations Standard Form G () s = τ s K + ζτs + 1 K = Gain τ = Natural Period of Oscillation ζ = Damping Factor (zeta) Note: this has to be 1.0!!! Corresponding Differential
More informationSolutions 2.4Page 140
Solutions.4Page 4 Problem 3 A mass of 3 kg is attached to the end of a spring that is stretched cm by a force of 5N. It is set in motion with initial position = and initial velocity v = m/s. Find the
More information24 th ITTC Benchmark Study on Numerical Prediction of Damage Ship Stability in Waves Preliminary Analysis of Results
th ITTC Benchmark Study on Numerical Prediction of Damage Ship Stability in Waves Preliminary Analysis of Results Apostolos Papanikolaou & Dimitris Spanos National Technical University of Athens Ship
More informationBIOMEDICAL ULTRASOUND
BIOMEDICAL ULTRASOUND Goals: To become familiar with: Ultrasound wave Wave propagation and Scattering Mechanisms of Tissue Damage Biomedical Ultrasound Transducers Biomedical Ultrasound Imaging Ultrasonic
More informationChapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.
Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems
More informationNUMERICAL SIMULATION OF REGULAR WAVES RUNUP OVER SLOPPING BEACH BY OPEN FOAM
NUMERICAL SIMULATION OF REGULAR WAVES RUNUP OVER SLOPPING BEACH BY OPEN FOAM Parviz Ghadimi 1*, Mohammad Ghandali 2, Mohammad Reza Ahmadi Balootaki 3 1*, 2, 3 Department of Marine Technology, Amirkabir
More informationSOLID MECHANICS DYNAMICS TUTORIAL MOMENT OF INERTIA. This work covers elements of the following syllabi.
SOLID MECHANICS DYNAMICS TUTOIAL MOMENT OF INETIA This work covers elements of the following syllabi. Parts of the Engineering Council Graduate Diploma Exam D5 Dynamics of Mechanical Systems Parts of the
More informationUnit  6 Vibrations of Two Degree of Freedom Systems
Unit  6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two
More informationExperiment #8, Error Analysis of the Period of a Simple Pendulum
Physics 181  Summer 013  Experiment #8 1 Experiment #8, Error Analysis of the Period of a Simple Pendulum 1 Purpose 1. To measure the period of a pendulum limited to small angular displacement. A pendulum
More informationGeneral Physics I Can Statements
General Physics I Can Statements Motion (Kinematics) 1. I can describe motion in terms of position (x), displacement (Δx), distance (d), speed (s), velocity (v), acceleration (a), and time (t). A. I can
More informationGauss Formulation of the gravitational forces
Chapter 1 Gauss Formulation of the gravitational forces 1.1 ome theoretical background We have seen in class the Newton s formulation of the gravitational law. Often it is interesting to describe a conservative
More informationBidirectional turbines for converting acoustic wave power into electricity
Bidirectional turbines for converting acoustic wave power into electricity Kees de BLOK 1, Pawel OWCZAREK 2, MauriceXavier FRANCOIS 3 1 Aster Thermoacoustics, Smeestraat 11, 8194LG Veessen, The Netherlands
More informationChapter 6 Circular Motion
Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example
More informationphysics 1/12/2016 Chapter 20 Lecture Chapter 20 Traveling Waves
Chapter 20 Lecture physics FOR SCIENTISTS AND ENGINEERS a strategic approach THIRD EDITION randall d. knight Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide
More informationDetailed simulation of mass spectra for quadrupole mass spectrometer systems
Detailed simulation of mass spectra for quadrupole mass spectrometer systems J. R. Gibson, a) S. Taylor, and J. H. Leck Department of Electrical Engineering and Electronics, The University of Liverpool,
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationPES 2130 Fall 2014, Spendier Lecture 10/Page 1
PES 2130 Fall 2014, Spendier Lecture 10/Page 1 Lecture today: Chapter 16 Waves1 1) Wave on String and Wave Equation 2) Speed of a transverse wave on a stretched string 3) Energy and power of a wave traveling
More informationMechanical Vibrations
Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME  TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More information